Download De finetti theorems, mean-field limits and bose

DE FINETTI THEOREMS, MEAN-FIELD LIMITS AND
BOSE-EINSTEIN CONDENSATION
arXiv:1506.05263v1 [math-ph] 17 Jun 2015
NICOLAS ROUGERIE
Laboratoire de Physique et Modélisation des Milieux Condensés, Université Grenoble 1 & CNRS.
Lecture notes for a course at the Ludwig-Maximilian Universität, Münich in April 2015.
Translated from a “Cours Peccot” at the collège de France, February-April 2014.
Abstract. These notes deal with the mean-field approximation for equilibrium states
of N -body systems in classical and quantum statistical mechanics. A general strategy
for the justification of effective models based on statistical independence assumptions is
presented in details. The main tools are structure theorems à la de Finetti, describing
the large N limits of admissible states for these systems. These rely on the symmetry
under exchange of particles, due to their indiscernability. Emphasis is put on quantum
aspects, in particular the mean-field approximation for the ground states of large bosonic
systems, in relation with the Bose-Einstein condensation phenomenon. Topics covered
in details include: the structure of reduced density matrices for large bosonic systems,
Fock-space localization methods, derivation of effective energy functionals of Hartree or
non-linear Schrödinger type, starting from the many-body Schrödinger Hamiltonian.
Date: June 2015.
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Dedicated to my daughter Céleste.
NICOLAS ROUGERIE
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
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Contents
Foreword
A few words about the experiments
Some mathematical questions raised by experiments
Plan of the course
Acknowledgements
1. Introduction: Problems and Formalism
1.1. Statistical mechanics and mean-field approximation
1.2. Quantum mechanics and Bose-Einstein condensation
1.3. Mean-field approximation and the classical de Finetti theorem
1.4. Bose-Einstein condensation and the quantum de Finetti theorem
2. Equilibrium statistical mechanics
2.1. The Hewitt-Savage Theorem
2.2. The Diaconis-Freedman theorem
2.3. Mean-field limit for a classical free-energy functional
2.4. Quantitative estimates in the mean-field/small temperature limit
3. The quantum de Finetti theorem and Hartree’s theory
3.1. Setting the stage
3.2. Confined systems and the strong de Finetti theorem
3.3. Systems with no bound states and the weak de Finetti theorem
3.4. Links between various structure theorems for bosonic states
4. The quantum de Finetti theorem in finite dimensonal spaces
4.1. The CKMR construction and Chiribella’s formula
4.2. Heuristics and motivation
4.3. Chiribella’s formula and anti-Wick quantization
5. Fock-space localization and applications
5.1. Weak convergence and localization for a two-body state
5.2. Fock-space localization
5.3. Proof of the weak quantum de Finetti theorem and corollaries
6. Derivation of Hartree’s theory: general case
6.1. The translation-invariant problem
6.2. Concluding the proof in the general case
7. Derivation of Gross-Pitaevskii functionals
7.1. Preliminary remarks
7.2. Statements and discussion
7.3. Quantitative estimates for Hartree’s theory
7.4. From Hartree to NLS
Appendix A. A quantum use of the classical theorem
A.1. Classical formulation of the quantum problem
A.2. Passing to the limit
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A.3. The limit problem
A.4. Conclusion
Appendix B. Finite-dimensional bosons at large temperature
B.1. Setting and results
B.2. Berezin-Lieb inequalities
B.3. Proof of Theorem B.1
References
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DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
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Foreword
The purpose of these notes is to present as exhaustively and pedagogically as possible
some recent mathematical results bearing on the Bose-Einstein condensation phenomenon
observed in ultra-cold atomic gases. One of the numerous theoretical problems posed by
these experiments is the understanding of the link between effective models, describing
the physics with a remarkable precision, and the first principles of quantum mechanics.
The process leading from the fundamental to the effective theories is often called a meanfield limit, and it has motivated a very large number of investigations in theoretical and
mathematical physics. In this course, we shall focus on one of the methods allowing to
deal with mean-field limits, which is based on de Finetti theorems. The emergence of the
mean-field models will be interpreted as a fundamental consequence of the structure of
physical states under consideration.
This text will touch on subjects from mathematical analysis, probability theory, condensed matter physics, ultra-cold atoms physics, quantum statistical mechanics and quantum information. The emphasis will be on the author’s speciality, namely the analytic
aspects of the derivation of mean-field equilibrium states. The presentation will thus have
a pronounced mathematical style, but readers should keep in mind the connection between the questions adressed here and cold atoms physics, in particular as regards the
experiments leading to the observation of Bose-Einstein condensates in the mid 90’s.
A few words about the experiments. Bose-Einstein condensation (BEC) is at the
heart of a rapidly expanding research field since the mid 90’s. The extreme versatility of
cold atoms experiments allows for a direct investigation of numerous questions of fundamental physics. For more thorough developments in this direction, I refer the reader to
the literature, in particular to [1, 18, 47, 49, 125, 152, 158, 72, 45] and references therein.
French readers will find very accessible discussions in [48, 37, 42].
The first experimental realizations of BEC took place at the MIT and at Boulder, in
the groups of W. Ketterle on the one hand and E. Cornell-C. Wieman on the other hand.
The 2001 Nobel prize in physics was awarded to Cornell-Wieman-Ketterle for this remarkable achievement. The possibilities opened up by these experiments for investigating
macroscopic quantum phenomena constitute a cornerstone of contemporary physics.
A Bose-Einstein condensate is made of a large number of particles (alkali atoms usually)
occupying the same quantum state. BEC thus requires that said particles be bosons, i.e.
that they do not satisfy Pauli’s exclusion principle which prevents mutliple occupancy of
a single quantum state.
This macroscopic occupancy of a unique low-energy quantum state appears only at very
low temperatures. There exists a critical temperature Tc for the existence of a condensate,
and macroscopic occupancy occurs only for temperatures T < Tc . The existence of such
a critical temperature was theoretically infered in works of Bose and Einstein [19, 63] in
the 1920’s. Important objections were however formulated:
(1) The critical temperature Tc is extremely low, unrealistically so as it seemed in
the 1920’s.
(2) At such a temperature, all known materials should form a solid state, and not be
gaseous as assumed in Bose and Einstein’s papers.
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(3) The argument of Bose and Einstein applies to an ideal gas, neglecting interactions
between particles, which is a serious drawback.
The first objection could be bypassed only in the 1990’s with the advent of powerful
techniques such as laser cooling1 and evaporative cooling. These allowed to reach temperatures in the micro-Kelvin range in quantum gases trapped by magneto-optics means.
As for the second objection, the answer has to do with the diluteness of the samples:
three-particles collisions necessary to initiate the formation of molecules, and ultimately
of a solid phase, are extremely rare in the experiments. One thus has the possibility of
observing a metastable gaseous phase during a sufficiently long time for a condensate to
form.
The third objection is of a more theoretical nature. Most of the material discussed in
these notes is part of a research program (of many authors, see references) whose goal is
to remove that objection. We will thus have the opportunity to discuss it a length in the
sequel.
Many agreeing observations have confirmed the experimental realization of BEC: imaging of the atoms’ distribution in momentum/energy space, interference of condensates,
superfluidity in trapped gases ... The importance thus acquired by the mathematical
models used in the description of this phenomenon has motivated a vast literature devoted to their derivation and analysis.
Some mathematical questions raised by experiments. In the presence of BEC,
the gas under consideration can be described by a single wave-functions ψ : Rd 7→ C,
corresponding to the quantum state in which all particles reside. A system of N quantum
particles should normally be described by a N -particle wave-function ΨN : RdN 7→ C.
One thus has to understand how and why can this huge simplification be justified, that is
how the collective behavior of the N particles emerges. The investigation of the precision
of this approximation, whose practical and theoretical consequences are fundamental, is a
task of extreme importance for theoretical and mathematical physicists.
One may ask the following questions:
(1) Can one describe the ground state (that is the equlibrium state at zero temperature) of an interacting Bose system with a single wave-function ψ ?
(2) Start from a single wave-function and let the system evolve along the natural
dynamics (N -body Schrödinger flow). Is the single wave-function description preserved by the dynamics ?
(3) Can one rigorously prove the existence of a critical temperature Tc under which the
finite temperature equilibrium states may be described by a single wave-function ?
These questions are three aspects of the third objection mentioned in the preceding
paragraph. We thus recall that the point is to understand the BEC phenomenon in the
presence of interactions. The case of an ideal gas is essentially trivial, at least for questions
1 and 2.
One should keep in mind that, in the spirit of statistical mechanics, we aim at justifying the single wave-function description asymptotically in the limit of large particle
numbers, under appropriate assumptions on the model under consideration. Ideally, the
assumptions should reduce to those ensuring that both the N -body model one starts from
11997 Nobel prize in physics: S. Chu-W. Phillips-C. Cohen-Tannoudji.
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
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and the 1-body model one arrives at are mathematically well-defined. Let us note that,
for interacting quantum particles, the former is always linear, while the latter is always
non-linear.
Many recent results presented in the sequel are generalizations to the quantum case
of better known results of classical statistical mechanics. Related questions indeed occur
also in this simpler context. For pedagogical reasons, some notions on mean-field limits
in classical mechanics will thus be recalled in these notes.
This course deals with question number 1, and we will be naturally lead to develop tools
of intrinsic mathematical interest. One may use essentially two approaches:
• The first one exploits particular properties of certain important physical models.
It thus applies differently to different models, and under often rather restrictive
assumptions, in particular as regards the shape of the inter-particle interactions.
A non-exhaustive list of references using such ideas is [173, 172, 167, 168, 179] for
the classical case, and [15, 133, 176, 86, 178, 125, 123] for the quantum case
• The second one is the object of these lectures. It exploits properties of the set
of admissible states, that is of N -body wave-functions ΨN . That we consider
bosonic particles implies a fundamental symmetry property for these functions.
This approach has the merit of being much more general than the first one, and
in many cases to get pretty close to the “ideally minimal assumptions” for the
validity of the mean-field approximation mentioned before.
A possible interpretation is to see the mean-field limit as a parameter regime
where correlations between particles become negligible. We will use very strongly
the key notion of bosonic symmetry. There will be many opportunities to discuss
the literature in details, but let us mention immediately [146, 31, 98, 99, 101, 170]
and [70, 71, 154, 159, 113, 115] for applications of these ideas in classical and
quantum mechanics respectively.
The distinction between the two philosophies is of course somewhat artificial since one will
often benefit from borrowing ideas to both, see e.g. [149] for a recent example.
To keep these notes reasonably short, question 2 will not be treated at all, although
a vast literature exists, see e.g. [92, 79, 80, 189, 10, 64, 65, 5, 69, 76, 165, 13, 105, 157]
and references therein, as well as the lecture notes [82, 14]. We note that the quantum de
Finetti theorems that we will discuss in the sequel have recently proved useful in dealing
with question 2, see [5, 6, 7, 36, 35]. The use of classical de Finetti theorems in a dynamic
framework is older [189, 190, 191]
As for question 3, this is a famous open problem in mathematical physics. Very little is
known at a satisfying level of mathematical rigor, but see however [177, 17]. We will touch
on questions raised by taking temperature into account only very briefly in Appendix B,
in a greatly simplified framework. This subject is further studied in the paper [114].
Plan of the course.
These notes are organized as follows:
• A long introduction, Chapter 1, recalls the basic formalism we shall need to give a
precise formulation to the problems we are interested in. We will start with classical
mechanics and then move on to quantum aspects. The question of justifying
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the mean-field approximation for equilibrium states of a given Hamiltonian will
be formulated in both contexts. The proof stategy that will occupy the bulk
of the notes will be described in a purely formal manner, so as to introduce as
quickly as possible the de Finetti theorems that will be our main tools. Similarities
between the classical and quantum frameworks are very strong. Differences show
up essentially when discussing the proofs of the fundamental de Finetti theorems.
• Chapter 2 is essentially independent from the rest of the notes. It contains the
analysis of classical systems: proof of the classical de Finetti theorem (also called
Hewitt-Savage theorem), application to equilibrium states of a classical Hamiltonian. The proof of the Hewitt-Savage theorem we will present, due to Diaconis and
Freedman, is purely classical and does not generalize to the quantum case.
• We start adressing quantum aspects in Chapter 3. Two versions (strong and
weak) of the quantum de Finetti theorem are given without proofs, along with
their direct applications to “relatively simple” bosonic systems in the mean-field
regime. Section 3.4 contains a discussion of the various versions of the quantum
de Finetti theorem and outlines the proof strategy that we shall follow.
• Chapters 4 and 5 contain the two main steps of the proof of the quantum de Finetti
theorem we choosed to present: respectively “explicit construction and estimates
in finite dimension” and “generalization to infinite dimensions via localization in
Fock space”. The proof should not be seen as a black box: not only the final result
but also the intermediary constructions will be of use in the sequel.
• Equiped with the results of the two previous chapters, we will be able to give in
Chapter 6 the justification of the mean-field approximation for the ground state of
an essentially generic bosonic system. Contrarily to the case treated in Chapter 3,
the quantum de Finetti theorem will not be sufficient in this case, and we will have
to make use of some of the ingredients introduced in Chapter 5.
• The mean-field limit is not the only physically relevant one. In Chapter 7 we will
study a dilute regime in which the range of the interactions goes to 0 when N → ∞.
In this case one obtains in the limit Gross-Pitaevskii (or non-linear Schrödinger)
functionals with local non-linearities. We will present a strategy for the derivation
of such objects based on the tools of Chapter 4.
The main body of the text is supplemented with two appendices containing each an
unpublished note of Mathieu Lewin and the author.
• Appendix A shows how, in some particular cases, one may use the classical de
Finetti theorem to deal with quantum problems. This strategy is less natural (and
less efficient) than that presented in Chapters 3 and 6, but it has a conceptual
interest.
• Appendix B deviates from the main line of the course since the Hilbert spaces under
consideration will be finite dimensional. In this context, combining a large temperature limit with a mean-field limit, one may obtain a theorem of semi-classical
nature which gives examples of de Finetti measures not encountered previously.
This will be the occasion to mention Berezin-Lieb inequalities and their link with
the considerations of Chapter 4.
Acknowledgements. The motivation to write the french version of these notes came
from the opportunity of presenting the material during a “cours Peccot” at the Collège
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
9
de France in the spring of 2014. Warm thanks to the audience of these lectures for their
interest and feedback, and to the Spartacus project (website: http://spartacus-idh.com)
for their proposal to publish the french version.
The motivation for translating the notes into english was provided by an invitation
to give an (expanded version of) the course at the LMU-Munich in the spring of 2015.
I thank in particular Thomas Østergaard Sørensen for the invitation and the practical
organization of the lectures, as well as Heinz Siedentop, Martin Fraas, Sergey Morozov,
Sven Bachmann and Peter Müller for making my stay in Münich very enjoyable. Feedback
from the students following the course was also very useful in revising the notes to produce
the current version.
I am of course indebted to my collaborators on the subjects of these notes: Mathieu
Lewin and Phan Thành Nam for the quantum aspects (more recently also Douglas Lundholm, Robert Seiringer and Dominique Spehner), Sylvia Serfaty and Jakob Yngvason for
the classical aspects. Fruitful exchanges with colleagues, in particular Zied Ammari, Francis Nier, Patrick Gérard, Isaac Kim, Jan-Philip Solovej, Jürg Fröhlich, Elliott Lieb, Eric
Carlen, Antti Knowles and Benjamin Schlein were also very useful.
Finally, I acknowledge financial support from the French ANR (Mathostaq project,
ANR-13-JS01-0005-01)
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1. Introduction: Problems and Formalism
Here we describe the mathematical objects that will be studied throughout the course.
Our main object of interest will be many-body quantum mechanics, but the analogy with
some questions of classical mechanics is instructive enough for us to also describe that
formalism. Questions of units and dimensionality are systematically ignored to simplify
notation.
1.1. Statistical mechanics and mean-field approximation. For pedagogical reasons
we shall recall some notions on mean-field limits in classical mechanics before going to the
quantum aspects, relevant to the BEC phenomenon. This paragraph aims at fixing notation and reviewing some basic concepts of statistical mechanics. We will limit ourselves
to the description of the equilibrium states of a classical system. Dynamic aspects are
voluntarily ignored, and the reader is refered to [82] for a review on these subjects.
Phase space. The state of a classical particle is entirely determined by its position x and
its speed v (or equivalently its momentum p). For a particle living in a domain Ω ⊂ Rd
we thus work in the phase space Ω × Rd , the set of possible positions and momenta. For
a N -particle system we work in ΩN × RdN .
Pure states. We call a pure state one where the positions and momenta of all particles are
known exactly. Equilibrium states at zero temperature for example are pure: in classical
mechanics, uncertainty on the state of a system is only due to “thermal noise”.
For a N -particle system, a pure state corresponds to a point
(X; P ) = (x1 , . . . , xN ; p1 , . . . , pN ) ∈ ΩN × RdN
in phase-space, where the pair (xi ; pi ) gives the position and momentum of particle number i. Having in mind the introduction of mixed states in the sequel, we will identify a
pure state with a superposition of Dirac masses
X
µX;P =
δXσ ;Pσ .
(1.1)
σ∈ΣN
This identification takes into account the fact that real particles are indistinguishable.
One can actually not attribute the pair (xi ; pi ) of position/momentum to any one of the
N particles in particular, whence the sum over the permutation group ΣN in (1.1). Here
and in the sequel our notation is
Xσ = (xσ(1) , . . . , xσ(N ) )
Pσ = (pσ(1) , . . . , pσ(N ) ).
(1.2)
Saying that the system is in the state µX;P means that one of the particles has position
and momentum (xi ; pi ), i = 1 . . . N , but one cannot specify which because of indistinguishability.
Mixed states. At non-zero temperature, that is when some thermal noise is present, one
cannot determine with certainty the state of the system. One in fact looks for a statistical
superposition of pure states, which corresponds to specifying the probability that the
system is in a certain pure state. One then speaks of mixed states, which are the convex
cominations of pure states, seen as Dirac masses as in Equation (1.1). The set of convex
combinations of pure states of course corresponds to the set of symmetric probability
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
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measures on phase-space. A general N -particles mixed state is thus a probability measure
µN ∈ Ps (ΩN × RdN ) satisfying
dµN (X; P ) = dµN (Xσ ; Pσ )
(1.3)
for all permutations σ ∈ ΣN . One interprets µN (X; P ) as the probability density that
particle i has position xi and momentum pi , i = 1 . . . N . Pure states of the form (1.1) are
a particular type of mixed states where the statistical uncertainty is reduced to zero, up
to indistinguishability.
Free energy. The energy of a classical system is specified by the choice of a Hamiltonian,
a function over phase-space. In non-relativistic mechanics, the kinetic energy of a particle
of momentum p is always m|p|2 /2. Taking m = 1 to simplify notation, we will consider
an energy of the form
HN (X; P ) :=
N
X
|pj |2
j=1
2
+
N
X
V (xj ) + λ
j=1
X
1≤i<j≤N
w(xi − xj )
(1.4)
where V is an external potential (e.g. electrostatic) felt by all particles and w a pairinteraction potential that we will assume symmetric,
w(−x) = w(x).
The real parameter λ sets the strength of interparticle interactions. We could of course
add three-particles, four-particles etc ... interactions, but this is seldom required by the
physics, and when it is, there is no additional conceptual difficulty.
The energy of a mixed state µN ∈ Ps (ΩN × RdN ) is then given by
Z
HN (X; P )dµ N (X; P )
(1.5)
E[µN ] :=
ΩN ×RdN
and by symmetry of the Hamiltonian this reduces to HN (X; P ) for a pure state of the
form (1.1). At zero temperature, equilibrium states are found by minimizing the energy
functional (1.5):
n
o
E(N ) = inf E[µN ], µN ∈ Ps (ΩN × RdN )
(1.6)
and the infimum (ground state energy) is of course equal to the minimum of the Hamiltonian HN . It is attained by a pure state (1.1) where (X; P ) is a minimum point for HN
(in particular P = (0, . . . , 0)).
In presence of thermal noise, one must take the entropy
Z
dµN (X; P ) log(µN (X; P ))
(1.7)
S[µN ] := −
ΩN ×RdN
into account. This is a measure of the degree of uncertainty on the state of the system.
Note for example that pure states have the lowest possible entropy: S[µN ] = −∞ if µN
if of the form (1.1). At temperature T , one finds the equilibrium state by minimizing the
free-energy functional
F[µN ] = E[µN ] − T S[µN ]
Z
Z
HN (X; P )dµN (X; P ) + T
=
ΩN ×RdN
ΩN ×RdN
dµN (X; P ) log(µN (X; P )) (1.8)
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which amounts to saying that the more probable states must find a balance between having
a low energy and having a large entropy. We shall denote
n
o
F (N ) = inf F[µN ], µN ∈ Ps (ΩN × RdN )
(1.9)
without specifying the temperature dependence. A minimizer must be a sufficiently regular
probability so that (minus) the entropy is finite
Momentum minimization. In the absence of a prescribed relation between the position and
the momentum distribution of a classical state, the minimization in momentum variables
of the above functionals is in fact trivial. A state minimizing (1.5) is always of the form
X
µN = δP =0 ⊗
δX=Xσ0
σ∈ΣN
where X 0 is a minimum point for HN (X; 0, . . . , 0), i.e. particles are all at rest. We are
thus reduced to looking for the minimum points of HN (X; 0, . . . , 0) as a function of X.
The minimization of (1.8) leads to a Gaussian in momentum variables mutliplied by a
Gibbs state in position variables2


N
X
1
1
1
1
2

|pj |
exp −
⊗
exp − HN (X; 0, . . . , 0) .
µN =
ZP
2T
ZN
T
j=1
Momentum variables thus no longer intervene in the minimization of the functionals determining equilibrium states and they will be completely ignored in the sequel. We will
keep the preceding notation for the minimization in position variables:
HN (X) =
N
X
V (xj ) + λ
j=1
E[µN ] =
F[µN ] =
Z
Z
ΩN
ΩN
X
1≤i<j≤N
w(xi − xj )
HN (X)dµN (X)
HN (X)dµN (X) + T
Z
ΩN
dµN (X) log(µN (X))
(1.10)
where µN ∈ Ps (ΩN ) is a symmetric probability measure in position variables only.
Marginals, reduced densities. Given a N -particles mixed state, it is very useful to consider
its marginals, or reduced densities, obtained by integrating out some variables:
Z
(n)
µ(x1 , . . . , xn , x′n+1 , . . . , x′N )dx′n+1 . . . dx′N ∈ Ps (Ωn ). (1.11)
µN (x1 , . . . , xn ) =
ΩN−n
(n)
The n−th reduced density µN is interpreted as the probability density for having one
particle at x1 , one particle at x2 , etc... one particle at xn . In view of the symmetry of
µN , the choice of which N − n variables over which one integrates in Definition (1.11) is
irrelevant.
2The partition functions Z and Z normalize the state in L1 .
P
N
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
13
A first use of these marginals is a rewriting of the energy using only the first two
marginals3:
ZZ
Z
N (N − 1)
(2)
(1)
w(x − y)dµN (x, y)
V (x)dµN (x) + λ
E[µN ] = N
2
Ω×Ω
Ω
ZZ
N
N
N (N − 1)
(2)
=
(1.12)
V (x) + V (y) + λ
w(x − y) dµN (x, y),
2
2
2
Ω×Ω
where we used the symmetry of the Hamiltonian.
Mean-field approximation. Solving the above minimization minimization problems is a
very difficult task when the particle number N gets large. In order to obtain more tractable
theories from which useful information can be extracted, one often has to rely on approximations. The simplest and most well-known of these is the mean-field approximation.
One can introduce it in several ways, the goal being to obtain a self-consistent one-body
problem starting from the N -body problem.
Here we follow the “molecular chaos” point of view on mean-field theory, originating in
the works of Maxwell and Boltzmann. The approximation consists in assuming that all
particles are independent and identically distributed (iid). We thus take an ansatz of the
form
N
Y
ρ(xj )
(1.13)
µN (x1 , . . . , xN ) = ρ⊗N (x1 , . . . , xN ) =
j=1
where ρ ∈ P(Ω) is a one-body probability density describing the typical behavior of one
of the iid particles under consideration.
The mean-field energy and free-energy functionals are obtained by inserting this ansatz
in (1.5) or (1.8). The mean-field energy functional is thus
ZZ
Z
N −1
w(x − y)dρ(x)dρ(y).
(1.14)
V (x)dρ(x) + λ
E MF [ρ] = N −1 E[ρ⊗N ] =
2
Ω×Ω
Ω
We shall denote E MF its infimum amongst probability measures. In a similar manner, the
mean-field free energy functional is given by
F MF [ρ] = N −1 F[ρ⊗N ]
Z
Z
ZZ
N −1
V (x)dρ(x) + λ
=
ρ log ρ
w(x − y)dρ(x)dρ(y) + T
2
Ω
Ω
Ω×Ω
(1.15)
and its infimum shall be denoted F MF . The term “mean-field” is motivated by the fact
that (1.14) corresponds to having an interaction between the particles’ density ρ and the
self-consistant potential
Z
w(. − y)dρ(y)
ρ∗w =
Ω
whose gradient is the so-called mean-field.
3More generally, an energy depending only on a n-body potential may be rewritten by using the n-th
marginal only.
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1.2. Quantum mechanics and Bose-Einstein condensation. After this classical intemezzo we can now introduce the quantum objects that are the main theme of this
course. We shall be content with a rapid review of the basic principles of quantum mechanics. Other “mathematician-friendly” presentations may be found in [124, 188]. Our
discussion of the relevant concepts is in places voluntarily simplified.
Wave-functions and quantum kinetic energy. One of the basic postulates of quantum mechanics is the identification of pure states of a system with normalized vectors of a complex
Hilbert space H. For particles living in the configuration space Rd , the relevant Hilbert
space is L2 (Rd ), the space of complex square-integrable functions over Rd .
Given a particle in the state ψ ∈ L2 (Rd ), one identifies |ψ|2 with a probability density:
|ψ(x)|2 gives the probability for the particle to be located at x. We thus impose the
normalization
Z
Rd
|ψ|2 = 1.
We thus see that, even in the case of a pure state, one cannot specify with certainty the
position of the particle. More precisely, one cannot simultaneously specify its position and
its momentum. This uncertainty principle is the direct consequence of another fundamental postulate: |ψ̂|2 gives the momentum-space probability density of the particle, where ψ̂
is the Fourier transform of ψ.
In quantum mechanics, the (non-relativistic) kinetic energy of a particle is thus given
by
Z
Z
1
|p|2
2
|ψ̂(p)| dp =
|∇ψ(x)|2 dx.
(1.16)
Rd 2
Rd 2
That the position and momentum of a particle cannot be simultaneously specified is a
consequence of the fact that it is impossible for both |ψ|2 and |ψ̂|2 to converge to a
Dirac mass. A popular way of quantifying this is Heisenberg’s uncertainty principle: for
all x0 ∈ Rd
Z
Z
2
2
2
|x − x0 | |ψ(x)| dx ≥ C.
|∇ψ(x)| dx
Rd
Rd
Indeed, the more precisely the particle’s position is known, the smaller the second term of
the left-hand side (for a certain x0 ). The first term of the left-hand side must then be very
large, which, in view of (1.16) rules out the possibility for the momentum distribution to
be concentrated around a single p0 ∈ Rd .
For many applications however (see [121] for a discussion of this point), this inequality is
not sufficient. A better way of quantifying the uncertainty principle is given by Sobolev’s
inequality (here in its 3D version):
Z
1/3
Z
6
2
|ψ|
.
|∇ψ| ≥ C
R3
R3
If the position of the particle is known with precision, |ψ|2 must approach a Dirac mass,
and the right-hand side of the above inequality blows up. So do the integrals (1.16), with
the same interpretation as previously.
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
15
d
Bosons and Fermions. For a system
NN 2 of dN quantum particles living in R , the appropriate
2
dN
L (R ). A pure state is thus a certain Ψ ∈ L2 (RdN )
Hilbert space is L (R ) ≃
2
where |Ψ(x1 , . . . , xN )| is interpreted as the probability density for having particle 1 at x1 ,
..., particle N at xN . As in classical mechanics, the indistinguishability of the particles
imposes
|Ψ(X)|2 = |Ψ(Xσ )|2
(1.17)
for any permutation σ ∈ ΣN . This condition is necessary for indistinguishable patricles,
but it is in fact not sufficient for quantum particles. To introduce the correct notion, we
denote Uσ the unitary operator interchanging particles according to σ ∈ ΣN :
Uσ u1 ⊗ . . . ⊗ uN = uσ(1) ⊗ . . . ⊗ uσ(N )
NN 2 d
L (R ) (one may
for all u1 , . . . , uN ∈ L2 (Rd ), extended by linearity to L2 (RdN ) ≃
construct a basis with vectors of the form u1 ⊗ . . . ⊗ uN ). For Ψ ∈ L2 (RdN ) to describe
indistinguishable particles we have to require that
hΨ, AΨiL2 (RdN ) = hUσ Ψ, AUσ ΨiL2 (RdN )
(1.18)
for any bounded operator A acting on L2 (RdN ). Details would lead us to far, but suffice it
to say that condition (1.18) corresponds to asking that any measure (corresponding to an
observable A) on the system must be independent of the particles’ labeling. In classical
mechanics, all possible measurements correspond to bounded functions on phase-space
and thus (1.3) alone guarantees the invariance of observations under particle exchanges.
In quantum mechanics, observables include all bounded operators on the ambient Hilbert
space, and one must thus impose the stronger condition (1.18).
An important consequence4 of the symmetry condition (1.18) is that a system of indistinguishable quantum particles must satisfy one of the following conditions, stronger
than (1.17): either
Ψ(X) = Ψ(Xσ )
(1.19)
for all X ∈ RdN and σ ∈ ΣN , or
Ψ(X) = ε(σ)Ψ(Xσ )
(1.20)
for all X ∈ RdN and σ ∈ ΣN , where ε(σ) is the sign of the permutation σ. One refers
to particles described by a wave-function satisfying (1.19) (respectively (1.20)) as bosons
(respectively fermions). These two types of fundamental particles have a very different
behavior, one speaks of bosonic and fermionic statistics. For example, fermions obey the
Pauli exclusion principle which stipulates that two fermions cannot simultaneously occupy
the same quantum state. One can already see that (1.20) imposes
Ψ(x1 , . . . , xi , . . . , xj , . . . , xN ) = −Ψ(x1 , . . . , xj , . . . , xi , . . . , xN )
and thus (formally)
Ψ(x1 , . . . , xi , . . . , xi , . . . , xN ) = 0
which implies that it is impossible for two fermions to occupy the same position xi . One
may consult [124] for a discussion of Lieb-Thirring inequalities, which are one of the most
important consequences of Pauli’s principle.
4This is an easy but non-trivial exercise.
16
NICOLAS ROUGERIE
Concretely, when studying a quantum system made of only one type of particles, one
has to restrict admissible pure states to those being either of bosonic of fermionic type.
One thus works
NN 2 d
• for bosons, in L2s (RdN ) ≃
s L (R ), the space of symmetric square-integrable
wave-functions, identified with the symmetric tensor product of N copies of L2 (Rd ).
NN 2 d
• for fermions, in L2as (RdN ) ≃
as L (R ), the space of anti-symmetric squareintegrable wave-functions, identified with the anti-symmetric tensor product of N
copies of L2 (Rd ).
As the name indicates, Bose-Einstein condensation can occur only in a bosonic system,
and this course shall thus focus on the first case.
Density matrices, mixed states. We will always identify a pure state Ψ ∈ L2 (RdN ) with
the corresponding density matrix, i.e. the orthogonal projector onto Ψ, denoted |Ψi hΨ|.
Just as in classical mechanics, mixed states of a system are by definition the statistical
superpositions of pure states, that is the convex combinations of orthogonal projectors.
Using the spectral theorem, it is clear that the set of mixed states conciı̈des with that of
positive self-adjoint operators having trace 1:
n
o
S(L2 (RdN )) = Γ ∈ S1 (L2 (RdN )), Γ = Γ∗ , Γ ≥ 0, Tr Γ = 1
(1.21)
where S1 (H) is the Schatten-von Neumann class [161, 174, 183] of trace-class operators on a
Hilbert space H. To obtain the bosonic (fermionic) mixed states one considers respectively
n
o
S(L2s/as (RdN )) = Γ ∈ S1 (L2s/as (RdN )), Γ = Γ∗ , Γ ≥ 0, Tr Γ = 1 .
Note that in the vocabulary of density matrices, bosonic symmetry consists in imposing
Uσ Γ = Γ
(1.22)
whereas fermionic symmetry corresponds to
Uσ Γ = ε(σ)Γ.
One sometimes considers the weaker symmetry notion
Uσ ΓUσ∗ = Γ
(1.23)
which is for example satisfied by the (unphysical) superposition of a bosonic and a
fermionic state.
Energy functionals. The quantum energy functional corresponding to the classical nonrelativistic Hamiltonian (1.4) is obtained by the substitution
p ↔ −i∇,
(1.24)
consistent with the identification (1.16) for the kinetic energy of a quantum particle. The
quantized Hamiltonian is a (unbounded) operator on L2 (RdN ):
N X
X
1
w(xi − xj )
(1.25)
HN =
− ∆j + V (xj ) + λ
2
j=1
1≤i<j≤N
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
17
where −∆j = (−i∇j )2 corresponds to the Laplacian in the variable xj ∈ Rd . The corresponding energy for a pure state Ψ ∈ L2 (RdN ) is
By linearity this generalizes to
E[Ψ] = hΨ, HN ΨiL2 (RdN ) .
(1.26)
E[Γ] = TrL2 (RdN ) [HN Γ].
(1.27)
in the case of a mixed state Γ ∈ S(L2 (RdN )). At zero temperature, the equilibrium state
of the system is found by minimizing the above energy functional. In view of the linearity
of (1.27) as a function of Γ and by the spectral theorem, it is clear that one may restrict
the minimization to pure states:
n
o
Es/as (N ) = inf E[Γ], Γ ∈ S(L2s/as (RdN ))
n
o
= inf E[Ψ], Ψ ∈ L2s/as (RdN ), kΨkL2 (RdN ) = 1 .
(1.28)
Here we use again the index s (respectively as) to denoted the bosonic (respectively
fermionic) energy. In the absence of an index, we mean that the minimization is performed
without symmetry constraint. However, because of the symmetry of the Hamiltonian, one
can in this case restrict the minimization to mixed states satisfying (1.23), or to wavefunctions satisfying (1.17).
In the presence of thermal noise, one must add to the energy a term including the
Von-Neumann entropy
X
aj log aj
(1.29)
S[Γ] = − TrL2 (RdN ) [Γ log Γ] = −
j
where the aj ’s are the eigenvalues (real and positive) of Γ, whose existence is guaranteed
by the spectral theorem. Similarly to the classical entropy (1.7), Von-Neumann’s entropy
is minimized (S[Γ] = 0) by pure states, i.e. orthongonal projectors, which of course only
have one non-zero eigenvalue, equal to 1.
The free-energy functional at temperature T is then
F[Γ] = E[Γ] − T S[Γ]
(1.30)
and minimizers are in general mixed states.
Alternative forms for the kinetic energy: magnetic fields and relativistic effects. In the
classical case, we have only introduced the simplest possible form of kinetic energy. The
reason is that for the minimization problems that shall concern us, the kinetic energy plays
no role. Other choices for the relation between p and the kinetic energy5 are nevertheless
possible. In quantum mechanics the choice of this relation is crucial even at equilibrium because of the non-trivial minimization in momentum variables. Recalling (1.16) and (1.24),
we also see that in the quantum context, different choices will lead to different functional
spaces in which to set the problem.
Apart from the non-relativistic quantum energy already introduced, at least two generalizations are physically interesting:
5Other dispersion relations.
18
NICOLAS ROUGERIE
• When a magnetic field B : Rd 7→ R interacts with the particles, one replaces
p ↔ −i∇ + A
(1.31)
where A : Rd 7→ Rd is the vector potential, satisfying
B = curl A.
Of course B determines A only up to a gradient. The choice of a particular A is
called a gauge choice. The kinetic energy operator, taking the Lorentz force into
account, becomes in this case
(p + A)2 = (−i∇ + A)2 = − (∇ − iA)2 ,
called a magnetic Laplacian.
This formalism is also appropriate for particles in a rotating frame: calling x3 the
rotation axis one must then take A = Ω(−x2 , x1 , 0) with Ω the rotation frequency.
This corresponds to taking the Coriolis force into account. In this case one must
also replace the potential V (x) by V (x) − Ω2 |x|2 to account for the centrifugal
force.
• When one wishes to take relativistic effects into account, the dispersion relation
becomes
p
Kinetic energy = c p2 + m2 c2 − mc2
with m the mass and c the speed of light in vacuum. Choosing units so that c = 1
and recalling (1.24), one is lead to consider the kinetic energy operator
p
p
p2 + m2 − m = −∆ + m2 − m
(1.32)
which is easily defined in Fourier variables for example. In the non-relativitic limit
where |p| ≪ m = mc we formally recover the operator −∆ to leading order. A
caricature of (1.32) is sometimes used, corresponding to the “extreme relativistic”
case |p| ≫ mc where one takes
p
√
p2 = −∆
(1.33)
as kinetic energy operator.
• One can of course combine the two generalizations to consider relativistic particles
in a magnetic field, using the operators
p
(−i∇ + A)2 + m2 − m and |−i∇ + A|
based on the relativistic dispersion relation and the correspondance (1.31).
Reduced density matrices. It will be useful to define in the quantum context a concept
similar to the reduced densities of a classical state. Given Γ ∈ S(L2 (RdN )), one defines
its n-th reduced density matrix by taking a partial trace on the last N − n particles:
Γ(n) = Trn+1→N Γ,
which precisely means that for any bounded operator An acting on
TrL2 (Rdn ) [An Γ(n) ] := TrL2 (RdN ) [An ⊗ 1⊗N −n Γ]
(1.34)
L2 (Rdn ),
where 1 is the identity on L2 (Rd ). The above definition is easily generalized to any
Hilbert space, but in the case of L2 , the reader is maybe more familiar with the following
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
19
equivalent definition: We identify Γ with its kernel, i.e. the function Γ(X; Y ) such that
for all Ψ ∈ L2 (RdN )
Z
Γ(X; Y )Ψ(Y )dY.
ΓΨ =
RdN
One can then also identify
Γ(n)
with its kernel
Γ(n) (x1 , . . . , xn ; y1 , . . . , yn )
Z
Γ(x1 , . . . , xn , zn+1 , . . . , zN ; y1 , . . . , yn , zn+1 , . . . , zN )dzn+1 . . . dzN .
=
Rd(N−n)
In the case where Γ has a symmetry, bosonic or fermionic, the choice of the variables over
which we take the partial trace is arbitrary. Let us note that even if one starts from a
pure state, the reduced density matrices are in general mixed states.
Just as in (1.12) one may rewrite the energy (1.27) in the form
h
i
N (N − 1)
E[Γ] = N TrL2 (Rd ) − 21 ∆ + V Γ(1) + λ
TrL2 (R2d ) [w(x − y)Γ(2) ]
2
N (N − 1)
N
w(x − y) Γ(2) .
− 21 ∆x + V (x) − 12 ∆y + V (y) + λ
= TrL2 (R2d )
2
2
(1.35)
Mean-field approximation. Even more than so in classical mechanics, actually solving the
minimization problem (1.28) for large N is way too costly, and it is necessary to rely on
approximations. This course aims at studying the simplest of those, which consists in
imitating (1.13) by taking an ansatz for iid particles,
Ψ(x1 , . . . , xN ) = u⊗N (x1 , . . . , xN ) = u(x1 ) . . . u(xN )
for a certain u ∈
Hartree functional
L2 (Rd ).
(1.36)
Inserting this in the energy functional (1.35) we obtain the
EH [u] = N −1 E[u⊗N ]
ZZ
Z N −1
1
|∇u|2 + V |u|2 + λ
|u(x)|2 w(x − y)|u(y)|2
=
2
2
d
d
d
R ×R
R
with the corresponding minimization problem
n
o
eH = inf EH [u], kukL2 (Rd ) = 1 .
(1.37)
(1.38)
Note that we have transformed a linear problem for the N -body wave-function (since the
energy functional is quadratic, the variational equation is linear) into a cubic problem for
the one-body wave-function u (the energy functional is quartic and hence the variational
equation will be cubic).
Ansatz (1.36) is a symmetric wave-function, appropriate for bosons. It corresponds to
looking for the ground state in the form of a Bose-Einstein condensate where all particles
are in the state u. Because of Pauli’s principle, fermions can in fact never be completely
uncorrelated in the sense of (1.36). The mean-field ansatz for fermions rather consists in
taking
Ψ(x1 , . . . , xN ) = det (uj (xk ))1≤j,k≤N
20
NICOLAS ROUGERIE
with orthonormal functions u1 , . . . , uN (orbitals of the system). This ansatz (Slater determinant) leads to the Hartree-Fock functional that we shall not discuss in these notes (a
presentation in the same spirit can be found in [166])
Note that for classical particles, the mean-field ansatz (1.13) allows taking a mixed
state ρ. To describe a bosonic system, one always takes a pure state u in the ansatz (1.36),
which calls for the following remarks:
• If one takes a general γ ∈ S(L2 (Rd )), the N -body state defined as
Γ = γ ⊗N
(1.39)
has the bosonic symmetry (1.22) if and only if γ is a pure state (see e.g. [94]),
γ = |uihu|, in which case Γ = |u⊗N ihu⊗N | and we are back to Ansatz (1.36).
• In the case of the energy functional (1.35), the minimization problems with and
without bosonic symmetry imposed coı̈ncide (see e.g. [124, Chapter 3]). One can
then minimize with no constraint and find the bosonic energy. This remains true
when the kinetic energy is (1.32) or (1.33) but is notoriously wrong in the presence
of a magnetic field.
• For some energy functionals, for example in presence of a magnetic or rotation
field, the minimum without bosonic symmetry is strictly lower than the minimum
with symmetry, cf [175]. The ansatz (1.39) is then appropriate to approximate
the problem without bosonic symmetry (in view of the Hamiltonian’s symmetry,
one may always assume the weaker symmetry (1.23)) and one then obtains a
generalized Hartree functional for mixed one-body states γ ∈ S(L2 (Rd )):
N −1
TrL2 (R2d ) w(x − y)γ ⊗2 .
(1.40)
EH [γ] = TrL2 (Rd ) − 12 ∆ + V γ + λ
2
The next two sections introduce, respectively in the classical and quantum setting, the
question that will occupy us in the rest of the course:
Can one justify, in a certain limit, the validity of the mean-field ansätze (1.13)
and (1.36) to describe equilibrium states of a system of indistinguishable particles ?
1.3. Mean-field approximation and the classical de Finetti theorem.
Is it legitimate to use the simplification (1.13) to determine the equilibrium states of a
classical system ? Experiments show this is the case when the number of particles is large,
which we mathematically implement by considering the limit N → ∞ of the problem at
hand.
A simple framework in which one can justify the validity of the mean-field approximation
is the so-called mean-fied limit, where one assumes that all terms in the energy (1.5) have
comparable weights. In view of (1.12), this is fulfilled if λ scales as N −1 , for example
λ=
1
,
N −1
(1.41)
in which case one may expect the ground state energy per particle N −1 E(N ) to have a
well-defined limit. The particular choice (1.41) helps in simplifying certain expressions,
but the following considerations apply as soon as λ is of order N −1 . We should insist on the
simplification we introduced in looking at the N → ∞ limit under the assumption (1.41).
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
21
This regime is far from covering all the physically relevant cases. It however already leads
to an already very non-trivial and instructive problem.
The goal of this section is to dicuss the mean-field approximation for the zerotemperature equilibrium states of a classical system. We are thus looking at
Z
N
HN dµN , µN ∈ Ps (Ω )
(1.42)
E(N ) = inf
ΩN
with
HN (X) =
N
X
j=1
V (xj ) +
1
N −1
X
1≤i<j≤N
w(xi − xj ).
(1.43)
We will sketch a formal proof of the validity of the mean-field approximation at the level
of the ground state energy, i.e. we shall argue that
E(N )
= E MF = inf E MF [ρ], ρ ∈ P(Ω)
N →∞ N
lim
(1.44)
where the mean-field functional is obtained as in (1.14), taking (1.41) into account:
Z
ZZ
1
V dρ +
E MF [ρ] =
w(x − y)dρ(x)dρ(y).
(1.45)
2 Ω×Ω
Ω
Estimate (1.44) for the ground state energy is a first step, and the proof scheme in fact
yields information on the equilibrium states themselves. To simplify the presentation, we
shall postpone the discussion of this aspect, as well as the study of the positive temperature
case, to Chapter 2.
Formally passing to the limit. Here we wish to make the “algebraic” structure of the
problem apparent. The justification of the manipulations we will perform requires analysis
assumptions that we shall discuss in the sequel of the course, but that we ignore for the
moment.
We start by using (1.12) and (1.41) to write
Z Z
1
E(N )
(2)
N
H2 (x, y)dµN (x, y), µN ∈ Ps (Ω )
= inf
N
2
Ω×Ω
(2)
where H2 is the two-body Hamiltonian defined as in (1.43) and µN is the second marginal
of the symmetric probability µN . Since the energy depends only on the second marginal,
one may see the problem we are interested in as a constrained optimization problem for
two-body symmetric probability measures:
Z Z
E(N )
1
(2)
(2)
(2)
H2 (x, y)dµ (x, y), µ ∈ PN
= inf
N
2
Ω×Ω
with
o
n
(2)
(2)
PN = µ(2) ∈ Ps (Ω2 ) | ∃µN ∈ Ps (ΩN ), µ(2) = µN
the set of two-body probability measures one can obtain as the marginals of a N -body
state. Assuming that the limit exists (first formal argument) we thus obtain
Z Z
1
E(N )
(2)
(2)
(2)
H2 (x, y)dµ (x, y), µ ∈ PN
=
lim inf
(1.46)
lim
N →∞ N
2 N →∞
Ω×Ω
22
NICOLAS ROUGERIE
and it is very tempting to exchange limit and infimum in this expression (second formal
argument) in order to obtain
Z Z
1
E(N )
(2)
(2)
(2)
H2 (x, y)dµ (x, y), µ ∈ lim PN .
= inf
lim
N →∞
N →∞ N
2
Ω×Ω
We have here observed that the energy functional appearing in (1.46) is actually independent of N . All the N -dependence lies in the constrained variational space we consider.
This suggests that a natural limit problem consists in minimizing the same functional but
on the limit of the variational space, as written above.
(2)
(2)
To give a meaning to the limit of PN , we observe that the sets PN form a decreasing
sequence
(2)
(2)
PN +1 ⊂ PN
(2)
as is easily shown by noting that if µ(2) ∈ PN +1 , then for a certain µN +1 ∈ Ps (ΩN +1 )
(N ) (2)
(2)
(1.47)
µ(2) = µN +1 = µN +1
(N )
and of course µN +1 ∈ Ps (ΩN ). One can thus legitimately identify
\ (2)
(2)
(2)
PN
:= lim PN =
P∞
N →∞
N ≥2
and the natural limit problem is then (modulo the justification of the above formal manipulations)
Z Z
1
E(N )
(2)
(2)
(2)
H2 (x, y)dµ (x, y), µ ∈ P∞ .
(1.48)
= E∞ = inf
lim
N →∞ N
2
Ω×Ω
This is a variational problem over the set of two-body states that one can obtain as reduced
densities of N -body states, for all N .
A structure theorem. We now explain that actually
E∞ = E MF ,
(2)
as a consequence of a fundamental result on the structure of the space P∞ .
Let us take a closer look at this space. Of course it contains the product states of the
(2)
form ρ ⊗ ρ, ρ ∈ P(Ω) since ρ⊗2 is the second marginal of ρ⊗N for all N . By convexity P∞
also contains all the convex combinations of product states:
)
(Z
(2)
(1.49)
ρ⊗2 dP (ρ), P ∈ P(P(Ω)) ⊂ P∞
ρ∈P(Ω)
with P(P(Ω)) the set of probability measures over P(Ω). In view of (1.12) and (1.45) we
will have justified the mean-field approximation (1.44) if one can show that the infimum
in (1.48) is attained for µ(2) = ρ⊗2 for a certain ρ ∈ P(Ω).
The structure result that allows us to reach this conclusion is the observation that there
is in fact equality in (1.49):
)
(Z
(2)
ρ⊗2 dP (ρ), P ∈ P(P(Ω)) = P∞
.
(1.50)
ρ∈P(Ω)
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
23
Indeed, in view of the the linearity of the energy functional as a function of µ(2) , one may
write
)
(Z
ZZ
1
⊗2
H2 (x, y)dρ (x, y)dP (ρ), P ∈ P(P(Ω))
E∞ = inf
2
ρ∈P(Ω)
Ω×Ω
)
(Z
E MF [ρ]dP (ρ), P ∈ P(P(Ω))
= inf
ρ∈P(Ω)
= E MF
since it is clear that the infimum over P ∈ P(P(Ω)) is attained for P = δ̺MF , a Dirac
mass at ̺MF , a minimizer of E MF .
We thus see that, if we accept some formal manipulations (that we will justify in Chapter 2) to arrive at (1.48), the validity of the mean-field approximation follows by using very
little of the properties of the Hamiltonian, but a lot the structure of symmetric N -body
states, in the form of the equality (1.50).
The latter is a consequence of the Hewitt-Savage, or classical de Finetti, theorem [50,
51, 93], recalled in Section 2.1 and proved in Section 2.2. Let us give a few details for the
familiarized reader. We have to show the reverse inclusion in (1.49). We thus pick some
µ(2) satisfying
(2)
µ(2) = µN
for a certain sequence µN ∈ Ps (ΩN ). Recalling (1.47) we may assume that
(N )
µN +1 = µN
and we thus have to deal with a sequence (hierarchy) of consistant N -body states. There
then exists (Kolmogorov’s extension theorem) a symmetric probability measure over the
sequences of Ω, µ ∈ Ps (ΩN ) such that
µN = µ(N )
where the N -th marginal is defined as in (1.11). The Hewitt-Savage theorem [93] then
ensures the existence of a unique probability measure P ∈ P(P(Ω)) such that
Z
ρ⊗N dP (ρ).
µN =
ρ∈P(Ω)
We obtain the desired result by taking the second marginal.
Chapter 2 contains the details of the above proof scheme. We will specify the adequate
assumptions to put all these considerations on a rigorours basis. It is however worth to
note immediately that this proof (inspired from [146, 98, 99, 31, 101]) used no structure
property of the Hamiltonian (e.g. the interactions can be attractive or repulsive or a
mixture of both) but only compactness and regularity properties.
1.4. Bose-Einstein condensation and the quantum de Finetti theorem.
We now sketch a strategy for justifying the mean-field approximation at the level of
the ground state energy of a large bosonic system. The method is similar to that for the
24
NICOLAS ROUGERIE
classical case presented above. We consider a mean-field regime by setting λ = (N − 1)−1
and thus work with the Hamiltonian
HN =
N
X
j=1
− (∇j + iA(xj ))2 + V (xj ) +
1
N −1
X
1≤i<j≤N
w(xi − xj )
(1.51)
acting on L2s (RdN ). Compared to the previous situation, we have added a vector potential
A to start emphasizing the generality of the approach. It can for example correspond to
an external magnetic field B = curl A.
The starting point is the bosonic ground state energy defined as previously
n
o
E(N ) = inf TrL2 (RdN ) [HN ΓN ] , ΓN ∈ S(L2s (RdN ))
n
o
= inf hΨN , HN ΨN i , ΨN ∈ L2s (RdN ), kΨN kL2 (RdN ) = 1
(1.52)
where we recall that we can minimize over pure or mixed states indifferently.
Our goal is to show that for large N the bosonic energy per particle can be calculated
using Hartree’s functional
lim
N →∞
E(N )
= eH
N
(1.53)
where
n
o
eH = inf EH [u], u ∈ L2 (Rd ), kukL2 (Rd ) = 1
ZZ
Z
1
|u(x)|2 w(x − y)|u(y)|2 dxdy.
EH [u] =
|(∇ + iA)u|2 + V |u|2 +
2
d
d
d
R ×R
R
(1.54)
Since Hartree’s energy is obtained by inserting an anzatz ΨN = u⊗N in the N -body
energy functional, the asymptotic result (1.53) is already a strong indication in favor of
the existence of BEC in the ground state of bosonic system, in the mean-field regime. We
will come back later to the consequences of (1.53) for minimizers. As in the classical case,
the mean-field regime is a very simplified, but already very instructive, model. We will
present in Chapter 7 an anlysis of other physically relevant regimes.
Formally passing to the limit. The first step to obtain (1.53) is as in the classical case to
reduce formally to a simplified limit problem. We start by rewriting the energy using (1.35)
and the assumption λ = (N − 1)−1
o
i
h
n
E(N )
1
(2)
(1.55)
= inf TrL2 (R2d ) H2 Γ(2) , Γ(2) ∈ PN
N
2
where
n
o
(2)
PN = Γ(2) ∈ S(L2s (R2d )) | ∃ ΓN ∈ S(L2s (RdN )), Γ(2) = Tr3→N [ΓN ]
is the set of ”N -representable” two-body density matrices that are the partial trace of a
N -body state.
(2)
Characterizing the set PN is a famous problem in quantum mechanics, see e.g. [43, 124],
and the formulation (1.55) is thus not particularly useful at fixed N . However, as in the
classical case, the representability problem can be given a satisfactory answer in the limit
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
25
N → ∞, and we shall rely on this fact. We start by noticing that, taking partial traces,
we easily obtain
(2)
(2)
(1.56)
PN +1 ⊂ PN
so that (1.55) can be seen as a variational problem set in a variational set that gets more
and more constrained when N increases.
Assuming again that one can exchange infimum and limit (which is of course purely
formal) we obtain
o
i
h
n
E(N )
1
(2)
(1.57)
lim
= E∞ := inf TrL2 (R2d ) H2 Γ(2) , Γ(2) ∈ P∞
N →∞ N
2
with
\ (2)
(2)
(2)
(1.58)
PN
P∞
= lim PN =
N →∞
N ≥2
the set of two-body density matrices that are N -representable for all N . As previously we
have used the fact that the energy functional does not depend on N to pass to the limit
formally.
(2)
A structure theorem. It so happens that the structure of the set P∞ is entirely known
and implies the equality
E∞ = eH ,
(1.59)
which concludes the proof of (1.53), up to the justification of the formal manipulations
we have just performed.
The structure property leading to (1.59) is a quantum version of the de Finetti-Hewitt(2)
Savage theorem mentioned in the prevous section. Pick a Γ(2) ∈ P∞ . We then have a
sequence ΓN ∈ S(L2s (RdN )) such that for all N
(2)
Γ(2) = ΓN .
Without loss of generality one can assume that this sequence is consistent in the sense
that
(N )
ΓN +1 = TrN +1 [ΓN +1 ] = ΓN .
The quantum de Finetti theorem of Størmer-Hudson-Moody [192, 94] then guarantees the
existence of a probability measure P ∈ P(SL2 (Rd )) on the unit sphere of L2 (Rd ) such
that
Z
|u⊗N ihu⊗N |dP (u)
ΓN =
u∈SL2 (Rd )
and thus
Γ
(2)
=
Z
u∈SL2 (Rd )
|u⊗2 ihu⊗2 |dP (u).
We can then conclude that
( Z
)
1
E∞ = inf
Tr 2 2d [H2 |u⊗2 ihu⊗2 |]dP (u), P ∈ P(SL2 (Rd ))
2 u∈SL2 (Rd ) L (R )
)
(Z
EH [u]dP (u), P ∈ P(SL2 (Rd ))
= inf
u∈SL2 (Rd )
= eH
26
NICOLAS ROUGERIE
where the last equality holds because it is clearly optimal to pick P = δuH with uH a
minimizer of Hartree’s functional.
We thus see that the validity of (1.53) is (at least formally) a consequence of the
structure of the set of bosonic states and only marginally depends on properties of the
Hamiltonian (1.51). The justification (under some assumptions) of the formal manipulations we just performed to arrive at (1.57) (or a variant), as well as the proof the quantum
de Finetti theorem (along with generalizations and corollaries) are the main purpose of
these notes.
Bose-Einstein condensation. Let us anticipate a bit on the implications of (1.53). We
shall see in the sequel that results of the sort
Z
(n)
|u⊗n ihu⊗n |dP (u)
(1.60)
ΓN →
u∈SL2 (Rd )
for all fixed n ∈ N when N → ∞, follow very naturally in good cases. Here ΓN is
(the density matrix of) a minimizer of the N -body energy and P a probability measure
concentrated on minimizers of EH . The convergence will take place (still in good cases) in
trace-class norm.
When there is a unique (up to a constant phase) minimizer uH for EH we thus get
(n)
⊗n
ΓN → |u⊗n
H ihuH | when N → ∞,
(1.61)
which proves BEC at the level of the ground state. Indeed, BEC means by definition
(see [125] and references therein, in particular [151]) the existence of an eigenvalue of
(1)
order6 1 in the limit N → ∞ in the spectrum of ΓN . This is clearly implied by (1.61),
which is in fact stronger.
One can certainly wonder whether stronger results than (1.61) may be obtained. One
could for example think of an approximation in norm like
ΨN − u⊗N
H 2 dN → 0.
L (R
)
Let us mention immediately that results of this kind are wrong in general. The good
condensation notion is formulated using density matrices, as the following two remarks
show:
• Let us think of a ΨN of the form (⊗s stands for the symmetric tensor product)
⊗(N −1)
ΨN = uH
⊗s ϕ
where ϕ is orthogonal to uH . Such a state is “almost condensed” since all particles
but one are in the state uH . However, in the usual L2 (RdN ) sense, ΨN is orthogonal
to u⊗N
H . The two states thus cannot be close in norm, although their n-body density
matrices for n ≪ N will be close.
• Following the same line of ideas, it is natural to look for corrections to the N -body
minimizer under the form
⊗(N −1)
ΨN = ϕ0 u⊗N
H + uH
⊗(N −2)
⊗s ϕ1 + uH
⊗s ϕ2 + . . . + ϕN
with ϕ0 ∈ C and ϕk ∈ L2s (Rdk ) for k = 1 . . . N . It so happens that the above ansatz
is correct if the sequence (ϕk )k=0,...,N is chosen to minimize a certain effective
6Let us recall that all our density matrices are normalized to have trace 1 in this course.
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
27
Hamiltonian on Fock space. The occurence of non-condensed terms implying the
ϕk for k ≥ 1 contributes to leading order to the norm of ΨN , but not to the reduced
density matrices, which rigorously confirms that the correct notion of condensation
is necessarily based on reduced density matrices. This remark only scratches the
surface of a beautiful subject that we will not discuss here: Bogoliubov’s theory.
We refer the interested reader to [46, 130, 131, 187, 176, 86, 117, 150, 56] for recent
mathematical results in this direction.
A remark concerning symmetry. We have been focusing on the bosonic problem because of
our main physical motivations. The fermionic problem is (at least at present) not covered
by such considerations, but one might be interested in the problem without symmetry
constraint mentioned previously.
In the case where A ≡ 0 in (1.51), the problems with and without bosonic symmetry
coı̈ncide, but it is not the case in general. One may nevertheless use the same method
as above to study the problem without symmetry constraint. Indeed, because of the
symmetry of the Hamiltonian, one may without loss of generality impose the weaker
symmetry notion (1.23) in the minimization problem. The set of two-body density matrices
appearing in the limit is also covered by the Størmer-Hudson-Moody theorem, and one
then has to minimize an energy amongst two-body density matrices of the form
Z
(2)
γ ⊗2 dP (γ)
(1.62)
Γ =
γ∈S(L2 (Rd ))
where P is now a probability measure on mixed one-particle states, that is on positive selfadjoint operators over L2 (Rd ) having trace 1. One deduces in this case the convergence of
the ground state energy per particle to the minimum of the generalized Hartree functional
(cf (1.40))
N −1
TrL2 (R2d ) w(x − y)γ ⊗2
EH [γ] = TrL2 (Rd ) −(∇ + iA)2 + V γ + λ
2
and the minimum is in general different from the minimum of Hartree’s functional (1.54)
(see e.g. [175]).
We already recalled that γ ⊗2 can have bosonic symmetry only if γ is a pure state
γ = |uihu|. it is thus clear why the problem without symmetry can in general lead to a
strictly lower energy. In fact the minimizer of (1.40) when there is no magnetic field is
always attained at a pure state, in coherence with the different observations we already
made on symmetry issues. We finally note that taking bosonic symmetry into account
in the mean-field limit is done completely naturally using the different versions of the
quantum de Finetti theorem.
28
NICOLAS ROUGERIE
2. Equilibrium statistical mechanics
In this chapter we shall prove the classical de Finetti theorem informally discussed above
and present applications to problems in classical mechanics.
To simplify the exposition and with a view to applicaions, we will consider particles
living in a domain Ω ⊂ Rd which could be Rd itself. We denote ΩN and ΩN the cartesian
product of N copies of Ω and the set of sequences of Ω respectively. The space of probability
measures over a set Λ will always be denoted P(Λ). In places we may make the simplifying
assumption that Ω is compact, in which case P(Ω) is compact for the weak convergence
of measures.
2.1. The Hewitt-Savage Theorem.
We mention only in passing the first works on what is now known as de Finetti’s
theorem [50, 51, 97, 58]. In this course we shall start from [93] where the classical de
Finetti theorem is proved in its most general form. Our point of view on the de Finetti
theorem is here resolutely analytic in that we shall only deal with sequences of probability
measures. More probabilistic versions of the theorem exist. We refer the reader to [2, 96]
and references therein for developments in this direction.
Informally, the Hewitt-Savage theorem [93] says that every symmetric probability measure over ΩN approaches a convex combination of product probability measures when N
gets large. A symmetric probability measures µN has to satisfy
µN (A1 × . . . × AN ) = µN (Aσ(1) , . . . , Aσ(N ) )
(2.1)
ρ⊗N (A1 , . . . , AN ) = ρ(A1 ) . . . ρ(AN )
(2.2)
for every borelian domains A1 , . . . , AN ⊂ Ω and every permutation of N indices σ ∈ ΣN .
We denote Ps (ΩN ) the set of such probability measures. A product measure built on
ρ ∈ P(Ω) is of the form
and is of course symmetric. We are thus looking for a result looking like
Z
ρ⊗N dPµN (ρ) when N → ∞
µN ≈
(2.3)
ρ∈P(Ω)
where PµN ∈ P(P(Ω)) is a probability measure over probability measures.
A first possibility for giving a meaning to (2.3) consists in taking immediately N = ∞,
that is, consider a probability with infinitely many variables µ ∈ P(ΩN ) instead of µN ,
which lives on ΩN . This is the natural meaning one should give to a “classical state
of a system with infinitely many particles”. We assume a notion of symmetry inherited
from (2.1):
µ(A1 , A2 , . . .) = µ(Aσ(1) , Aσ(2) , . . .)
(2.4)
for every sequence of borelian domains (Ak )k∈N ⊂ ΩN and every permutation of infinitely many indices σ ∈ Σ∞ . We denote Ps (ΩN ) the set of probability measures over ΩN
satisfying (2.4). Hewitt and Savage [93] proved the following:
Theorem 2.1 (Hewitt-Savage 1955).
Let µ ∈ Ps (ΩN ) satisfy (2.4). Let µ(n) be its n-th marginal,
µ(n) (A1 , . . . , An ) = µ(A1 , . . . , An , Ω, . . . , Ω, . . .).
(2.5)
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
There exists a unique probability measure Pµ ∈ P(P(Ω)) such that
Z
(n)
ρ⊗n dPµ (ρ).
µ =
29
(2.6)
ρ∈P(Ω)
In statistical mechanics, this theorem is applied as a weak version of the formal approximation (2.3) in the following manner. We start from a classical N -particle state
µN ∈ P(ΩN ) whose marginals
(n)
µN (A1 , . . . , An ) = µ(A1 , . . . , An , ΩN −n )
(2.7)
converge7 as measures in P(Ωn ), up to a (non-relabeled) subsequence:
(n)
This means that
µN ⇀∗ µ(n) ∈ P(Ω(n) ).
(2.8)
(n)
µN (An ) → µ(n) (An )
for all borelian subset An of Ωn , and thus
Z
Z
(n)
fn dµ(n)
fn dµN →
Ω
Ω
for all bounded continuous functions decaying at infinity fn from Ωn to R. Modulo a
diagonal extraction argument one may always assume that the convergence (2.8) is along
the same subsequence for any n ∈ N. Testing (2.8) with a Borelian An = Am × Ωm−n for
m ≤ n one obtains the consistency relation
(m)
µ(n)
= µ(m) , for all m ≤ n
(2.9)
which implies that (µ(n) )n∈N does describe a system with infinitely many particles. One
may then see (Kolmogorov’s extension theorem) that there exists µ ∈ P(ΩN ) such that
µ(n) is its n-th marginal (whence the notation). This measure satisfies (2.4) and we can
thus apply Theorem 2.1 to obtain
Z
(n)
ρ⊗n dPµ (ρ) when N → ∞
(2.10)
µ N ⇀∗
ρ∈P(Ω)
where Pµ ∈ P(P(Ω)). This is a weak but rigorous version of (2.3). In words the nth marginal of a classical N -particle state may be approximated by a convex
combination of product states when N gets large and n is fixed . Note that the
measure Pµ appearing in (2.10) does not depend on n.
Of course, one must first be able to use a compactness argument in order to obtain (2.8).
This is possible if Ω is compact (then P(Ωn ) is compact for the convergence in the sense of
measures (2.8)). More generally, if the physical problem we are interested in has a confining
mechanism, one may show that the marginals of equilibrium states of the system are tight
and deduce (2.8).
As for the proof of Theorem 2.1, there are several possible approaches. We mention
first that the uniqueness part is a rather simple consequence of a density argument in
Cb (P(Ω)), the bounded continuous functions over P(Ω) due to Pierre-Louis Lions [140].
7We denote this convergence ⇀ to distinguish it from a norm convergence.
∗
30
NICOLAS ROUGERIE
Proof of Theorem 2.1, Uniqueness. We easily verify that monomials of the form
Z
Cb (P(Ω)) ∋ Mk,φ (ρ) := φ(x1 , . . . , xk )dρ⊗k (x1 , . . . , xk ), k ∈ N, φ ∈ Cb (Ωk )
(2.11)
generate a dense sub-algebra of Cb (P(Ω)), the space of bounded continuous functions over
P(Ω), see Section 1.7.3 in [82]. The main point is that for all µ ∈ P(Ω)
Mk,φ (µ)Mℓ,ψ (µ) = Mk+ℓ,φ ⊗ ψ (µ).
That this sub-algbra is dense is a consequence of the Stone-Weierstrass theorem.
We thus need only check that if there exists two measures Pµ and Pµ′ satisfying (2.5),
then
Z
Z
Mk,φ (ρ)dPµ (ρ) =
ρ∈P(Ω)
ρ∈P(Ω)
Mk,φ (ρ)dPµ′ (ρ)
for all k ∈ N and φ ∈ Cb (P(Ωk )). But this last equation simply means that
Z
Z
⊗k
φ(x1 , . . . , xk )dρ (x1 , . . . , xk ) dPµ (ρ)
ρ∈P(Ω)
Ωk
=
Z
ρ∈P(Ω)
Z
Ωk
φ(x1 , . . . , xk )dρ
⊗k
(x1 , . . . , xk ) dPµ′ (ρ)
which is obvious since both expressions are equal to
Z
φ(x1 , . . . , xk )dµ(k) (x1 , . . . , xk )
Ωk
by assumption.
For the existence of the measure, which is the most remarkable point, we mention three
approaches
• The original proof of Hewitt-Savage is geometrical: the set of symmetric probability
measures over ΩN is of course convex. The Choquet-Krein-Milman theorem says
that any point of a convex set is a convex combination of the extremal points. It
thus suffices to show that the extremal points of Ps (ΩN ) coı̈ncide with product
measures, corresponding to sequence of marginals of the form (ρ⊗n )n∈N . This
approach is not constructive, and the proof that the sequences (ρ⊗n )n∈N are the
extremal points of Ps (ΩN ) is by contradiction.
• An entirely constructive approach is due to Diaconis-Freedman [57]. In this probabilistic argument, Theorem 2.1 is a corollary of an approximation result at finite
N , giving a quantitative version of (2.10).
• Lions developed a new approach for the needs of mean-field game theory [140].
This is a dual point of view where one starts from “weakly dependent continuous
functions” with many variables. A summary of this is in the lecture notes [82,
Section 1.7.3], and a thorough presentation in [147, Chapter I]. Developments and
generalizations may be found in [91].
It so happens that the proof of the Hewitt-Savage theorem following Lions’ point of view
is for a large part a rediscovery of the method of Diaconis and Freedman. In the sequel
we will follow a blend of the two appraoches. See [147] for a more complete discussion.
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
31
2.2. The Diaconis-Freedman theorem.
As we just announced, it is in fact possible to give a quantitative version of (2.10)
which implies Theorem 2.1. Apart from its intrinsic interest this result naturally leads to
a constructive proof of the measure existence. The approximation will be quantified in
the natural norm for probability measures over a set S, the total variation norm:
Z
Z
d|µ| = sup φ dµ
(2.12)
kµkTV =
S
φ∈Cb (S)
Ω
which coı̈ncides with the L1 for absolutely continuous measures. We shall prove the
following result, taken from [57]:
Theorem 2.2 (Diaconis-Freedman).
Let µN ∈ Ps (ΩN ) be a symmetric probability measure. There exists PµN ∈ P(P(Ω)) such
that, setting
Z
ρ⊗N dPµN (ρ)
(2.13)
µ̃N :=
ρ∈P(Ω)
we have
(n)
(n) µN − µ̃N TV
≤2
n(n − 1)
.
N
(2.14)
Proof. We slightly abuse notation by writing µN (Z)dZ instead of dµN (Z) for integrals in
(z1 , . . . , zN ) = Z ∈ ΩN . It is anyway already instructive enough to consider the case of an
absolutely continuous measure.
By symmetry of µN we have, for all X = (x1 , . . . , xN ) ∈ ΩN ,
Z
X
µN (Z)
(N !)−1 δX=Zσ dZ
(2.15)
µN (X) =
ΩN
σ∈ΣN
where Zσ is the N -tuple (zσ(1) , . . . , zσ(n) ). We define
Z
X
µN (Z)
N −N δX=Zγ dZ,
µ̃N (X) =
ΩN
(2.16)
γ∈ΓN
where ΓN is the set of all applications 8 from {1, . . . , N } to itself and Xγ is defined is the
same manner as Xσ . The precise meaning of (2.16) is
Z
Z
X
−N
φ(X)dµ̃N (X) =
N
φ(Zγ )dµN (Z)
ΩN
ΩN
γ∈ΓN
for every regular function φ.
Noting that
X
γ∈ΓN

N −N δX=Zγ = N −1
N
X
j=1
8Compared to Σ , Γ thus allows repeated indices.
N
N
⊗N
δzj 
(x1 , . . . , xN ) ,
(2.17)
32
NICOLAS ROUGERIE
one can put (2.16) under the form (2.13) by taking
Z
N
X
N −1 δzj =x .
δρ=ρ̄Z µN (Z)dZ, ρ̄Z (x) :=
PµN (ρ) =
ΩN
(2.18)
i=1
Note in passing that PµN charges only empirical measures (of the form ρ̄Z above). We
now estimate the difference between the marginals of µN and µ̃N . Diaconis and Freedman
proceed as follows: of course

(n) 
(n) 
Z
X
 X

(n)
(n)
(N !)−1 δX=Zσ  − 
µN − µ̃N =
N −N δX=Zγ   µN (Z)dZ,

ΩN
σ∈ΣN
but
γ∈ΓN


X
σ∈ΣN
(n)
(N !)−1 δX=Zσ 
is the probability law for drowing n balls at random from an urn containing N balls9,
without replacement, whereas
(n)

X

N −N δX=Zγ 
γ∈ΓN
is the probability law for drawing n balls at random from an urn containing N , with
replacement. Intuitively it is clear that when n is small compared to N , the fact that we
replace the balls or not after each drawing does not significantly influence the result. It is
not difficult to obtain quantitative bounds leading to (2.14), see for example [75].
Another way of obtaining (2.14), which seems to originate in [87], is as follows: in view
of (2.16) and (2.17), we have
⊗n

Z
N
X
(n)
δzj  (x1 , . . . , xN ) dZ.
µN (Z) N −1
µ̃N (X) =
ΩN
j=1
We then expand the tensor product and compute the contribution of terms where all
(n)
indices are different. By symmetry of µN we obtain
N (N − 1) . . . (N − n + 1) (n)
(2.19)
µN + νn
Nn
where νn is a positive measure on Ωn (all terms obtained by expanding (2.17) are positive).
We thus have
N (N − 1) . . . (N − n + 1)
(n)
(n)
(n)
µN − µ̃N = 1 −
(2.20)
µ N − νn
Nn
and since both terms in the left-hand side are probabilities, we deduce
Z
N (N − 1) . . . (N − n + 1)
.
dνn = 1 −
Nn
Ωn
(n)
µ̃N =
9Labeled z , . . . , z .
1
N
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
33
Moreover, since the first term of the right-hand side of (2.20) is positive and the second
one negative we obtain by the triangle inequality
Z
Z
N (N − 1) . . . (N − n + 1)
(n)
(n) dνn
+
d µN − µ̃N ≤ 1 −
Nn
Ωn
Ωn
N (N − 1) . . . (N − n + 1)
=2 1−
.
Nn
It is then easy to see that
n
n j−1
N (N − 1) . . . (N − n + 1) Y N − j + 1 Y
1−
=
=
Nn
N
N
j=1
j=1
n(n − 1)
n−1 n
≥1−
,
≥ 1−
N
N
which proves (2.14) with C = 2. A better constant C = 1 can be obtained, see [57, 75]. The following remark can be useful in applications:
Remark 2.3 (First marginals of the Diaconis-Freedman measure).
We have
(1)
(1)
µ̃N (x) = µN (x)
(2.21)
and
N − 1 (2)
1 (1)
µN (x1 , x2 ) + µN (x1 )δx1 =x2 .
N
N
as direct consequences of Definition (2.17). Indeed, using symmetry,
(2)
µ̃N (x1 , x2 ) =
(1)
µ̃N (x) = N −1
N Z
X
j=1
(2)
µ̃N (x1 , x2 ) = N −2
= N −2
Z
ΩN
(1)
ΩN
+N
X
N Z
X
i=1
=
µN (Z)δzj =x dZ = µN (x)

µN (Z) 
1≤i6=j≤N
−2
(2.22)
ΩN
Z
ΩN
N
X
j=1

δzj =x1  
N
X
j=1

δzj =x2  dZ
µN (Z)δzi =x1 δzj =x2 dZ
µN (Z)δzi =x1 δzi =x2 dZ
1 (1)
N − 1 (2)
µN (x1 , x2 ) + µN (x1 )δx1 =x2 .
N
N
Higher-order marginals can be obtained by similar but heavier computations.
(2.23)
As a corollary of the preceding theorem we obtain a simple proof of the existence part
in the Hewitt-Savage theorem:
34
NICOLAS ROUGERIE
Proof of Theorem 2.1, Existence. We start with the case where Ω is compact. We apply
Theorem 2.2 to µ(N ) , obtaining
Z
n2
(n)
ρ⊗n dPN (ρ) ≤ C
(2.24)
µ −
N
ρ∈P(Ω)
TV
with PN ∈ P(P(Ω)) the measure defined in (2.18). When Ω is compact, P(Ω) and P(P(Ω))
also are. We thus may (up to a subsequence) assume that
PN → P ∈ P(P(Ω))
in the sense of measures and there only remains to pass to the limit N → ∞ at fixed n
in (2.24).
When Ω is not compact, we follow an idea of Lions [140] (see also [82]). We want to
ensure that the measure PN obtained by applying the Diaconis-Freedman theorem to µ(N )
converges. For this it suffices to test against a monomial of the form (2.11):


Z
Z
N
X
1
δzj  dµ(N ) (Z)
Mk,φ 
Mk,φ (µ)dPN (µ) =
N
Z∈ΩN
P(P(Ω))
j=1


Z
Z
k
N
Y
X
1
φ(x1 , . . . , xk )
=
δzj =xk  dµ(N ) (Z)
N
Z∈ΩN X∈Ωk
j=1
j=1
Z
φ(z1 , . . . , zk )dµ(k) (Z) + O(N −1 )
(2.25)
=
Z∈Ωk
by a computation similar to that yielding (2.19). The limit (2.25) thus exists for any
monomial Mk,φ , and by density of monomials for any bounded continuous function over
P(Ω). We then deduce
PN ⇀∗ P
and we can conclude as previously.
Some remarks before going to the applications of Theorems 2.1 and 2.2:
Remark 2.4 (On the Diaconis-Freedman-Lions construction).
(1) We first note that the measure defined by (2.18) is that Lions uses in his approach
of the Hewitt-Savage theorem. Using empirical measures is the canonical way to
construct a measure over P(P(Ω)) given one over Ps (ΩN ). Passing to the limit
as in (2.25) can replace the explicit estimate (2.14) if one is only interested in the
proof of Theorem 2.1. This construction and the combinatorial trick (2.19) are
also used e.g. in [83, 148, 91].
(2) Having an explicit estimate of the form (2.14) at hand is very satisfying and can
prove useful in applications. One might wonder whether the obtained convergence
rate is optimal. Perhaps suprisingly it √is. One might have expected a useful
estimate for n ≪ N , but it happens that n ≪ N is optimal, see examples in [57].
(3) The formulae (2.23) are useful in practice (see [170] for an application). It is quite
(2)
(2)
(1)
(1)
satisfying that µN = µ̃N and that µN can be reconstructed using only µ̃N
(1)
and µ̃N .
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
35
(4) As a drawback of its generality, the previous construction actually behaves very
badly in many cases. Note that (2.18) charges only empirical measures, which
all have infinite entropy. This causes problems when employing Theorem 2.2 to
the study of a functional with temperature. Moreover, in a situation with strong
repulsive interactions, one typically applies the construction to a measure with
(2)
zero probability of having two particles at the same place, µN (x, x) ≡ 0. In this
(1)
(2)
case µ̃N (x, x) = N −1 µN (x) is non-zero and the energy of µ̃N will be infinite if
the interaction potential has a singularity at the origin.
(5) It would be very interesting to have a construction leading to an estimate of the
form (2.14), bypassing the aforementioned inconveniences. For example, is it possible to guarantee that µ̃N ∈ L1 (ΩN ) if µN ∈ L1 (ΩN ) ? One might also demand
that the construction leave product measures ρ⊗N invariant, which is not at all
the case for (2.13).
(6) If Ω is replaced by a finite set, say Ω = {1, . . . , d}, one may obtain an error
proportional to dn/N instead of n2 /N by using the original proof of DiaconisFreedman. One may thus replace (2.14) by
C
(n)
(n) min dn, n2 .
(2.26)
µN − µ̃N ≤
N
TV
We will not use this point anywhere in the sequel.
Finally, let us make a separate remark on a possible generalization of the approach
above:
Remark 2.5 (Weakly dependent symmetric functions of many variables.).
In applications (see next section), one is lead to apply de Finetti-like theorems in the
following manner: given a sequence (uN )N ∈N of symmetric functions of N variables, we
study the quantity
Z
uN (X)dµN (X)
ΩN
for a symmetric probability measure µN . The results previously discussed imply that if
(n)
this happens to depend only on a marginal µN for fixed n, a natual limit object appears
when N → ∞. In particular, if
−1
X
N
φ(xi1 , . . . , xin ),
(2.27)
uN (X) =
n
1≤i1 <...<in ≤N
we have
Z
Z
uN (X)dµN (X) =
ΩN
Ωn
(n)
φ(x1 , . . . , xn )dµN
→
Z
ρ∈P(Ω)
Z
φ dρ
Ωn
⊗n
dPµ (ρ)
(2.28)
for a certain probability measure Pµ ∈ P(P(Ω)). One might ask whether this kind of
results is true for a class of functions uN depending on N in more subtle a manner. The
natural assumptions seems to be that uN depends weakly on its N variables, in the sense
introduced by Lions [140] and recalled in [82, Section 1.7.3]. Without entering the details,
one may easily see that to such a sequence there corresponds (modulo extraction of a
36
NICOLAS ROUGERIE
subsequence) a continuous function over probabilities U ∈ C(P(Ω)): one may show that,
along a subsequence,
Z
Z
U (ρ)dPµ (ρ).
(2.29)
uN (X)dµN (X) →
ΩN
ρ∈P(Ω)
Think of the case where uN depends only on the empirical measure:


N
X
1
δxj 
uN (x1 , . . . , xN ) = F 
N
(2.30)
j=1
with a sufficiently regular function F . Such a function depends weakly on its N variables
in the sense of Lions, but cannot be written in the form (2.27). More generally, this kind
of considerations can be applied under assumptions of the kind
C
∀j = 1 . . . N, ∀(x1 , . . . , xN ) ∈ ΩN .
|∇xj uN (x1 , . . . , xN )| ≤
N
One may then wonder whether the convergence rate in (2.29) can be quantified. Results
in this direction may be found in [91].
2.3. Mean-field limit for a classical free-energy functional.
In this section we apply Theorem 2.1 to the study of a free-energy functional at positive
temperature, following [146, 31, 99, 101]. We consider a domain Ω ⊂ Rd and the functional
Z
Z
µ(X) log µ(X)dX
(2.31)
HN (X)µ(X)dX + T
FN [µ] =
ΩN
X∈ΩN
P(ΩN ).
defined for probability measures µ ∈
Here the temperature T will be fixed in the
limit N → ∞ and the Hamiltonian HN is in mean-field scaling:
HN (X) =
N
X
j=1
V (xj ) +
1
N −1
X
1≤i<j≤N
w(xi − xj ).
(2.32)
We denote by V a lower semi-continuous potential. To be in a compact setting we shall
assume that either Ω is bounded or
V (x) → ∞ when |x| → ∞.
(2.33)
The interaction potential w will be bounded below and lower semi-continuous. To be
concrete, one may think of w ∈ L∞ , or a repulsive Coulomb potential:
1
w(x) =
if d = 3
(2.34)
|x|d−2
w(x) = − log |x| if d = 2
(2.35)
w(x) = −|x| if d = 1,
(2.36)
ubiquitous in applications. We will always take w even,
w(x) = w(−x),
and if the domain is not bounded we will assume
w(x − y) + V (x) + V (y) → ∞ when |x| → ∞ or |y| → ∞
w(x − y) + V (x) + V (y) is lower semi-continuous.
(2.37)
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
37
We are interested in the limit of the Gibbs measure minimizing (2.31) in Ps (ΩN ):
1
1
(2.38)
exp − HN (X) dX
µN (X) =
ZN
T
and to the corresponding free-energy
FN =
inf
µ∈P(ΩN
s )
FN [µ] = FN [µN ] = −T log ZN .
(2.39)
The free-energy functional is rewritten
Z
Z
ZZ
N
(1)
(2)
V (x)dµ (x) +
FN [µ] = N
µ log µ
w(x − y)dµ (x, y) + T
2
Ω
ΩN
Ω×Ω
Z
ZZ
N
(2)
µ log µ.
(2.40)
(w(x − y) + V (x) + V (y)) dµ (x, y) + T
=
2
ΩN
Ω×Ω
using the marginals
(n)
µ
(x1 , . . . , xn ) =
Z
dµ(x1 , . . . , xN ).
(2.41)
xn+1 ,...,xN ∈Ω
Inserting an ansatz of the form
µ = ρ⊗N , ρ ∈ P(Ω)
(2.42)
in (2.31) we obtain the mean-field functional
F MF [ρ] := N −1 FN [ρ⊗N ]
Z
Z
ZZ
1
V (x)dρ(x) +
=
ρ log ρ (2.43)
w(x − y)dρ(x)dρ(y) + T
2
Ω
Ω
Ω×Ω
with minimum F MF and minimizer (not necessarily unique) ̺MF amongst probability
measures. Our goal is to justify the mean-field approximation by proving the following
theorem:
Theorem 2.6 (Mean-field limit at fixed temperature).
We have
FN
→ F MF when N → ∞.
N
Moreover, up to a subsequence, we have for every n ∈ N
Z
(n)
ρ⊗n dP (ρ).
µ N ⇀∗
(2.44)
(2.45)
ρ∈MMF
as measures where P is a probability measurer over MMF , the set of minimizers of F MF .
In particular, if F MF has a unique minimizer we obtain, for the whole sequence,
⊗n
(n)
.
µN ⇀∗ ̺MF
Proof. We follow [146, 31, 98, 99]. An upper bound to the free energy is eeasily obtained
using test functions of the form ρ⊗N and we deduce
FN
≤ F MF .
N
(2.46)
38
NICOLAS ROUGERIE
The corresponding lower bound requires more work. We start by extracting a subsequence
along which
(n)
µN ⇀∗ µ(n)
(2.47)
for all n ∈ N, with µ ∈ Ps (ΩN ). This is done as explained in Section 2.1, using either
the fact that Ω is compact or Assumption (2.37), which implies that the sequence is tight.
Indeed, using simple energy upper and lower bounds one easily sees that for all ε > 0
there exists Rε such that
(2)
µN (B(0, Rε ) × B(0, Rε )) > 1 − ε
for all N . Tightness of all the other marginals follows.
By lower semi-continuity we immediately have
ZZ
1
(2)
(w(x − y) + V (x) + V (y)) dµN (x, y)
lim inf
N →∞ 2
Ω×Ω
ZZ
1
≥
(w(x − y) + V (x) + V (y)) dµ(2) (x, y). (2.48)
2 Ω×Ω
For the entropy term we use the sub-additivity property (this is a consequence of Jensen’s
inequality, see [163] or the previously cited references)
Z
Z
Z
N
(N −n⌊ Nn ⌋)
(N −n⌊ Nn ⌋)
(n)
(n)
µN log µN ≥
log µN
µN log µN + N−n N µN
⌊n⌋
n
ΩN
Ω
Ωn
where ⌊ . ⌋ denotes the integer part. Jensen’s inequality implies that for probability measures µ and ν, the relative entropy of µ with respect to ν is positive:
Z
Z
Z
Z
µ
µ
µ
µ
µ
ν
log
ν
= 0.
µ log = ν log ≥
ν
ν
ν
ν
ν
We deduce that for all ν0 ∈ P(Ω)
Z


(N −n⌊ Nn ⌋)
µ
(N −n⌊ ⌋)
(N −n⌊ ⌋)
(N −n⌊ ⌋)

log µN
= N−n N µN
log  N
µN
N−n⌊ N
⊗(N −n⌊ N
⌊
⌋
⌋
n
n
n ⌋)
Ω
Ω
ν0
Z
N
⊗(N −n⌊ N
(N −n⌊ n ⌋)
n ⌋)
log ν0
+ N−n N µN
⌊
⌋
n
Ω
Z
N
(1)
≥ N −n
µN log ν0 .
n
Ω
N
n
N
n
Z
N
n
Choosing ν0 ∈ P(Ω) of the form ν0 = c0 exp(−c1 V ) it is not difficult to see that the last
integral is bounded below independently of N and we thus obtain that for all n ∈ N
Z
Z
1
1
lim inf
µN log µN ≥
µ(n) log µ(n)
(2.49)
N →∞ N ΩN
n Ωn
by lower semi-continuity of (minus) the entropy.
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
39
Gathering (2.48) and (2.49) we obtain a lower bound in terms of a functional of µ:
1
1
lim inf FN [µN ] ≥ F[µ] :=
N →∞ N
2
ZZ
(w(x − y) + V (x) + V (y)) dµ(2) (x, y)
Ω×Ω
Z
1
µ(n) log µ(n) . (2.50)
+ T lim sup
n
n
n→∞
Ω
The second term in the right-hand side is called (minus) the mean entropy of µ ∈ P(ΩN ).
We will next apply the Hewitt-Savage theorem to µ. The first term in F is obviously
affine as a function of µ, which is perfect to apply (2.6). One might however worry at the
sight of the second term which rather looks convex. In fact a simple argument of [163]
shows that this mean entropy is affine. It is a part of the statement that the lim sup is in
fact a limit:
Lemma 2.7 (Properties of the mean entropy).
The functional
Z
Z
1
1
(n)
(n)
µ log µ = lim
µ(n) log µ(n)
sup
n→∞ n Ωn
n→∞ n Ωn
(2.51)
is affine over P(ΩN ).
Proof. We start by proving that the sup equals the limit. Denote the former by S and
pick j such that
Z
Z
1
1
µ(j) log µ(j) = sup
µ(n) log µ(n) − εj = S − εj .
j Ωj
n
n
n→∞
Ω
Next let k ≥ j and write (Euclidean division)
k = mj + r with 0 ≤ r < j.
Then, by the subbaditivity property already mentioned
Z
Z
Z
1
m
1
(k)
(k)
(j)
(j)
µ log µ ≥
µ log µ +
µ(r) log µ(r)
k Ωk
k Ωj
k Ωr
Cj
mj
≥
(S − εj ) −
k
k
where Cj only depends on j. We then pass to the limit k → ∞ at fixed j. Noting that
then m → k/j we obtain
lim inf
k→∞
1
k
Z
Ωk
µ(k) log µ(k) ≥ S − εj .
Passing finally to the limit j → ∞ along an appropriate subsequence we can make εj → 0,
which proves the equality in (2.51).
40
NICOLAS ROUGERIE
To prove that the functional is affine, let µ1 , µ2 ∈ P(ΩN ) be given. We use the convexity
of x 7→ x log x and the monotonicity of x 7→ log x to obtain
Z Z
Z
1 (n) 1 (n)
1 (n) 1 (n)
1
1
(n)
(n)
(n)
(n)
µ log µ1 +
µ log µ2 ≥
µ + µ2
µ + µ2
log
2 Ωn 1
2 Ωn 2
2 1
2
2 1
2
Ωn
Z
Z
1
1
(n)
(n)
(n)
(n)
µ log µ1 +
µ log µ2
≥
2 Ωn 1
2 Ωn 2
Z
Z
log(2)
(n)
(n)
µ2
−
µ1 +
2
Ωn
Ωn
Z
Z
1
1
(n)
(n)
(n)
(n)
=
µ1 log µ1 +
µ log µ2 − log(2).
2 Ωn
2 Ωn 2
Dividing by n and passing to the limit we deduce that the functional is indeed affine. Thus F[µ] is affine. There remains to use (2.6), which gives a probability Pµ ∈ P(P(Ω))
such that
Z
1
lim inf FN [µN ] ≥
F[ρ⊗∞ ]dPµ (ρ)
N →∞ N
ρ∈P(Ω)
Z
F MF [ρ]dPµ (ρ) ≥ F MF .
=
ρ∈P(Ω)
ρ⊗∞
Here we denote
the probability over P(ΩN ) which has ρ⊗n for n-th marginal for all
n and we have used the fact that Pµ has integral 1. This concludes the proof of (2.44)
and (2.45) follows easily. It is indeed clear in view of the previous inequalities that Pµ
must be concentrated on the set of minimizers of the mean-field free-energy functional. 2.4. Quantitative estimates in the mean-field/small temperature limit.
Here we give an example where the Diaconis-Freedman construction is useful to supplement the use of the Hewitt-Savage theorem. As mentioned in Remark 2.4, this construction
behaves rather badly with respect to the entropy, but there is a fair number of interesting
problems where it makes sense to consider a small temperature in the limit N → ∞, in
which case entropy plays a small role.
A central example is that of log-gases. It is well-known that the distribution of eigenvalues of certain random matrices ensembles is given by the Gibbs measure of a classical
gas with logarithmic interactions. Moreover, it so happens that the relevant limit for
large matrices is a mean-field regime with temperature of the order of N −1 . Consider the
following Hamiltonian (assumptions on V are as previously, taking w = − log | . |):
HN (X) =
N
X
j=1
V (xj ) −
1
N −1
X
1≤i<j≤N
log |xi − xj |
(2.52)
where X = (x1 , . . . , xN ) ∈ RdN . The associated Gibbs measure
µN (X) =
1
exp (−βN HN (X)) dX
ZN
(2.53)
corresponds (modulo a β-dependent change of scale) to the distribution of of the eigenvalues of a random matrix in the following cases:
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
41
2
• d = 1, β = 1, 2, 4 and V (x) = |x|2 . We respectively obtain the gaussian real
symmetric, complex hermitian and quarternionic self-dual matrices.
2
• d = 2, β = 2 and V (x) = |x|2 . We then obtain the gaussian matrices without
symmetry conditon, the so-called Ginibre ensemble [78].
In these notes we give no further precisions on the random matrix aspect, and will simply
take the previous facts as a sufficient motivation to study the N → ∞ limit of the measures (2.53) with β fixed. This corresponds (compare with (2.38)) to taking T = β −1 N −1 ,
i.e. a very small temperature. For an introduction to random matrices and log-gases we
refer to [8, 73, 145]. For precise studies of the measures (2.53) following different methods
than what we shall do here, one may consult e.g. [11, 12, 26, 20, 21, 20, 34, 167, 173, 172].
In the case d = 2, measures of the form (2.53) also have a natural application to the
study of certain quantum wave-functions appearing in the study of the fractional quantum
Hall effect (see [168, 169, 170, 171] and references therein). Here too it is sensible to
consider β as being fixed.
The singularity at the origin of the logarithm poses difficulties in the proof, as indicated in Remark 2.4, but one may bypass them easily, contrarily to those linked to the
entropy. The following method is not limited to log-gases and can be re-employed in
various contexts.
Again, (2.53) minimizes a free-energy functional
Z
Z
1
µ(X) log µ(X)dX
(2.54)
HN (X)µ(X)dX +
FN [µ] =
βN ΩN
X∈ΩN
whose minimum we denote FN . The natural limit object is this time an energy functional
with no entropy term:
ZZ
Z
1
MF
log |x − y|dρ(x)dρ(y),
(2.55)
V dρ −
E [ρ] :=
2
Rd ×Rd
Rd
obtained by inserting the ansatz ρ⊗N in (2.54) and neglecting the entropy term, which is
manifestly of lower order for fixed β. We denote E MF and ̺MF respectively the minimum
energy and the minimizer (unique in this case by strict convexity of the functional). It is
well-known (see the previous references as well as [101]) that
1
(2.56)
N −1 FN = − log ZN → E MF when N → ∞
β
and
⊗n
(n)
.
(2.57)
µN ⇀∗ ̺MF
We shall prove (2.57) and give a quantitative version of (2.56), taking inspiration
from [170]:
Theorem 2.8 (Free-energy estimate for a log-gas).
For all β ∈ R, we have
1
E MF − CN −1 β −1 + log N + 1 ≤ − log ZN ≤ E MF + Cβ −1 N −1 .
β
(2.58)
Fine estimates for the partition function ZN of the log-gase seem to have become available only recently [109, 153, 167, 173, 172]. Amongst other things, the previous references
indicate that the correction to E MF is exactly of order N −1 log N .
42
NICOLAS ROUGERIE
Proof. We shall not elaborate on the upper bound, which is easily obtained by taking
the usual product ansatz and estimating the entropy. For the lower bound we shall use
Theorem 2.2. We first need a crude bound on the entropy term: by positivity of the
relative entropy (via Jensen’s inequality)
Z
µ
µ log ≥ 0 for all µ, ν ∈ P(ΩN )
ν
dN
R
one may write, using the probability measure
ν N = (c0 exp (−V (|x|)))⊗N ,
the lower bound
Z
RdN
µN log µN ≥
Z
RdN
µN log ν N = −N
Z
(1)
Rd
V dµN − N log c0 .
(2.59)
To obtain a lower bound to the energy we first need to regularize the interaction potential:
Let α > 0 be a small parameter to be optimized over later and
(
2
− log α + 12 1 − |z|
if |z| ≤ α
α2
− logα |z| =
(2.60)
− log |z|
if |z| ≥ α.
Clearly − logα |z| ≤ − log |z| is regular at the origin. Moreover, we have

|z|2
 1
d
si |z| ≤ α
− + 3
−
logα |z| =
α
α
0
dα
si |z| ≥ α.
(2.61)
Using the lower bound − logα |z| ≤ − log |z| to obtain
ZZ
Z
Z
N
(2)
(1)
logα |x − y|dµN (x, y)
V dµN −
HN (X)µ(X)dX ≥ N
2
d
d
d
N
R ×R
R
X∈Ω
we are now in a position where we can apply Theorem 2.2. More precisely we use the
explicit formulae (2.23):
Z
HN (X)µN (X)dX ≥
X∈ΩN
Z
ZZ
N2
(1)
(2)
V dµ̃N −
N
logα |x − y|dµN (x, y) + C logα (0). (2.62)
2(N − 1)
Rd
Rd ×Rd
Combining (2.59), (2.13) and recalling that the temperature T is equal to (βN )−1 we
obtain
Z
−1
EαMF [ρ]dPµN (ρ)
N FN [µN ] ≥
ρ∈P(Rd )
+ CN −1 logα (0) − β −1 ≥ EαMF − CN −1 log(α) − C(βN )−1 (2.63)
where EαMF is the minimum (amongst probability measures) of the modified functional
ZZ
Z
N
−1 −1
MF
logα |x − y|dρ(x)dρ(y).
V (1 − β N )dρ −
Eα [ρ] :=
2(N − 1)
Rd ×Rd
Rd
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
43
Exploiting the associated variational equation, (2.61) and the Feynman-Hellmann principle, it is not difficult to see that for small enough α
MF
Eα − E MF ≤ Cαd + CN −1
and we thus conclude
N −1 FN [µN ] ≥ E MF − CN −1 log(α) − CβN −1 − CN −1 − Cαd − C,
which gives the desired lower bound after optimizing over α (take α = N −1/d ).
Remark 2.9 (Possible extensions).
(1) One can also prove quantitative versions of (2.57) following essentially the above
method of proof. We shall not elaborate on this point for which we refer to [170].
(2) Another case we could deal with along the same lines is that of unitary, orthogonal
and symplectic gaussian random matrices ensembles introduced by Dyson [60, 61,
62]. In this case Rd is replaced by the unit circle, β = 1, 2, 4, V ≡ 0 in (2.52). In
this case, dividing the interaction by N − 1 is irrelevant.
44
NICOLAS ROUGERIE
3. The quantum de Finetti theorem and Hartree’s theory
Now we enter the heart of the matter, i.e. mean-field limits for large bosonic systems. We present first the derivation of the ground state of Hartree’s theory for confined
particles, e.g. living in a bounded domain. In such a case, the result is a rather straightforward consequence of the quantum de Finetti theorem proved by Størmer and HudsonMoody [192, 94]. The latter describes all the strong limits (in the sense of the trace-class
norm) of the reduced density matrices of a large bosonic system.
We will then move on to the more complex case of non-confined systems. In this
chapter we will assume that the interaction potential has no bound state (it could be
purely repulsive for example), the general case being dealt with later (Chapter 6). In
the absence of bound states it is sufficient to have at our disposal a de Finetti theorem
describing all the weak limits (in the sense of the weak-∗ topology on S1 ) of the reduced
density matrices.
The weak de Finetti theorem (introduced in [113]) implies the strong de Finetti theorem
and in fact the two results can be deduced from an even more general theorem appearing
in [192, 94]. In these notes I chose not to follow this approach, but rather that of [5,
113] which is more constructive. This will be discussed in details in Section 3.4, which
announces the plan of the next chapters.
3.1. Setting the stage.
To simplify the discusstion, we will focus on the case of non relativistic quantum particles, in the absence of a magnetic field. The Hamiltonian will thus have the general
form
N
X
X
1
w(xi − xj ),
(3.1)
Tj +
HN =
N −1
j=1
1≤i<j≤N
NN
acting on the Hilbert space HN
s =
s H, i.e. the symmetric tensor product of N copies of
H where H denotes the space L2 (Ω) for Ω ⊂ Rd . The operator T is a Schrödinger operator
T = −∆ + V
(3.2)
with V : Ω 7→ R and Tj acts on the j-th variable:
Tj ψ1 ⊗ . . . ⊗ ψN = ψ1 ⊗ . . . ⊗ Tj ψj ⊗ . . . ⊗ ψN .
We assume that T is self-adjoint and bounded below, and that the interaction potential
w : R 7→ R is bounded relatively to T (as operators): for some 0 ≤ β− , β+ < 1
− β− (T1 + T2 ) − C ≤ w(x1 − x2 ) ≤ β+ (T1 + T2 ) + C.
(3.3)
We also take w symmetric
w(−x) = w(x),
and decaying at infinity
w ∈ Lp (Ω) + L∞ (Ω), max(1, d/2) < p < ∞ → 0, w(x) → 0 when |x| → ∞.
(3.4)
We will always make an abuse of notation by writing w for the multiplication operator
by w(x1 − x2 ) on L2 (Ω).
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
45
Remark 3.1 (Checking assumptions on w).
One may check that assumptions (3.4) imply operator bounds of the form (3.3), using
standard functional inequalities. We quickly sketch the method here for completness.
Suppose that w ∈ Lp (Rd ), such that the conjugate Hölder exponent q is smaller than half
the critical exponent for the Sobolev injection H 1 (Rd ) ֒→ Lq (Rd ):
q=
We decompose |w| as
p∗
p
< ,
p−1
2
p∗ =
2d
.
d−2
(3.5)
|w| = |w|1|w|≤R + |w|1|w|≥R .
Then, pick f ∈ Cc∞ (R2d ) and write
Z Z
2
|hf, wf i| = w(x1 − x2 )|f (x1 , x2 )| dx1 dx2 d
d
Z Z R ×R
ZZ
w(x1 − x2 )1|w|≥R |f (x1 , x2 )|2 dx1 dx2 + R
≤
Rd ×Rd
Next, using Hölder’s and Sobolev’s inequalities:
ZZ
w(x1 − x2 )1|w|≥R |f (x1 , x2 )|2 dx1 dx2
Rd ×Rd
≤
Z
Rd
Z
p
Rd
|w(x1 − x2 )| 1|w|≥R dx2
Rd ×Rd
1/p Z
|f (x1 , x2 )|2 dx1 dx2 .
1/q
2q
|f (x1 , x2 )| dx2
dx1
Z
kf (., x2 )k2H 1 dx2
≤ w1|w|≥R Lp
Rd
Rd
with
p−1
+
q −1
= 1. All in all we thus have
ZZ
ZZ
2
|∇x1 f (x1 , x2 )| dx1 dx2 + R
|hf, wf i| ≤ w1|w|≥R Lp
Rd ×Rd
Rd
Since w ∈ Lp , w1|w|≥R Lp → 0 when R → ∞ and we thus have
|f (x1 , x2 )|2 dx1 dx2 .
|hf, wf i| ≤ h(R) |hf, −∆x1 f i| + (R + h(R)) hf, f i ,
where h(R) can be made arbitrarily small by taking R large. In particular we can take
h(R) < 1, and this leads to an operator inequality of the desired form (3.3).
The above assumptions ensure by well-known methods that HN is self-adjoint on the
domain of the Laplacian, and bounded below, see [160]. Our goal is to describe the ground
state of (3.1), i.e. a state achieving10
E(N ) = inf σHN HN =
inf
Ψ∈HN ,kΨk=1
hΨ, HN ΨiHN .
(3.6)
In the mean-field regime that concerns us here, we expect that any ground state satifies
ΨN ≈ u⊗N when N → ∞
(3.7)
10The ground state might not exist, in which case we think of a sequence of states asymptotically
achieving the infimum.
46
NICOLAS ROUGERIE
in a sense to be made precise later, which naturally leads us to the Hartree functional
1
EH [u] = N −1 u⊗N , HN u⊗N HN = hu, T uiH + hu ⊗ u, w u ⊗ uiH2s
2
ZZ
Z
1
|u(x)|2 w(x − y)|u(y)|2 dxdy.
(3.8)
|∇u|2 + V |u|2 +
=
2 Ω×Ω
Ω
We shall denote eH and uH the minimum and a minimizer for EH respectively. By the
variational principle we of course have the upper bound
E(N )
≤ eH
(3.9)
N
and we aim at proving a matching lower bound to obtain
E(N )
→ eH when N → ∞.
N
(3.10)
Remark 3.2 (Generalizations).
All the main ideas can be introduced in the preceding framework, we refer to [113] for
a discussion of generalizations. One can even think of the case where both V and w are
smooth compactly supported functions if one wishes to understand the method in the
simplest possible case.
A very interesting generalization consists in substituting the Laplacian in (3.2) by a
relativistic kinetic energy operator and/or including a magnetic field as described in Section 1.2. One must then adapt the assumptions (3.3) and (3.4) but the message stays the
same: the approach applies as long as the many-body Hamiltonian and the limit functional
are both well-defined.
Another possible generalization is the inclusion of interactions involving more than two
particles at a time, to obtain functionals with higher-order non-linearities in the limit. It
is of course necessary to consider a Hamiltonian in mean-field scaling by adding terms of
the form e.g.
X
w(xi − xj , xi − xk )
λN
1≤i<j,k≤N
N −2
with λN ∝
when N → ∞. It is also possible to take into account a more general
form than just the multiplication by a potential, under assumptions of the same type
as (3.3).
3.2. Confined systems and the strong de Finetti theorem.
By “confined system” we mean that we are dealing with a compact setting. We will
make one the two following assumptions: either
or
Ω ⊂ Rd is a bounded set
Ω = Rd and V (x) → ∞ when |x| → ∞
with V the potential appearing in (3.2). We will also assumte that
(3.11)
(3.12)
V ∈ Lploc (Ω), max(1, d/2) < p ≤ ∞.
In both cases it is well-known [161] that
T = −∆ + V has compact resolvent,
(3.13)
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
47
which allows to easily obtain strong convergence of the reduced density matrices of a
ground state of (3.1). One may then really think of the limit object as a quantum state
with infinitely many particles. Taking inspiration from the classical setting discussed
previously, the natural definition is the following:
Definition 3.3 (Bosonic state with infinitely many particles).
Let H be a complex separable Hilbert space and, for all n ∈ N, let Hns be the corresponding
bosonic n-particles space. We call a bosonic state with infinitely many particles a sequence
(γ (n) )n∈N of trace-class operators satisfying
• γ (n) is a bosonic n-particles state: γ (n) ∈ S1 (Hns ) is self-adjoint, positive and
TrHns [γ (n) ] = 1.
(3.14)
• the sequence (γ (n) )n∈N is consistent:
Trn+1 [γ (n+1) ] = γ (n)
where Trn+1 is the partial trace with respect to the last variable in
(3.15)
Hn+1 .
The key property is the consistency (3.15), which ensures that the sequence under
consideration does describe a physical state. Note that γ (0) is just a real number and that
consistency implies that TrHn [γ (n) ] = 1 for all n as soon as γ (0) = 1.
A particular case of symmetric state is a product state:
Definition 3.4 (Product state with infinitely many particles).
A product state with infinitely many particles is a sequence of trace-class operators γ (n) ∈
S1 (Hns ) with
γ (n) = γ ⊗n ,
(3.16)
for all n ≥ 0 where γ is a one-particle state. A bosonic product state is necessarily of the
form
γ (n) = (|uihu|)⊗n = |u⊗n ihu⊗n |
(3.17)
with u ∈ SH, the unit sphere of the one-body Hilbert space.
That bosonic product states are all of the form (3.17) comes from the observation
that if γ ∈ S1 (H) is not pure (i.e. is not a projector), then γ ⊗2 cannot have bosonic
symmetry [94].
The strong de Finetti theorem is the appropriate tool to describe these objects and
specify the link between the two previous definitions. In the following form, it is due to
Hudson and Moody [94]:
Theorem 3.5 (Strong quantum de Finetti).
Let H be a separable Hilbert space and (γ (n) )n∈N a bosonic state with infinitely many
particles on H. There exists a unique Borel probability measure µ ∈ P(SH) on the sphere
SH = {u ∈ H, kuk = 1} of H, invariant under the action11 of S 1 , such that
Z
|u⊗n ihu⊗n | dµ(u)
(3.18)
γ (n) =
SH
for all n ≥ 0.
11Multiplication by a constant phase eiθ , θ ∈ R.
48
NICOLAS ROUGERIE
In other words, every bosonic state with infinitely many particles is a convex
combination of bosonic product states. To deduce the validity of the mean-field
approximation at the level of the ground state of (3.1), it then suffices to show that the
limit problem is set in the space of states with infinitely many particles, which is relatively
easy in a compact setting. We are going to prove the following result:
Theorem 3.6 (Derivation of Hartree’s theory for confined bosons).
Under the preceding assumptions, in particular (3.11) or (3.12)
lim
N →∞
E(N )
= eH .
N
Let ΨN be a ground state for HN , achieving the infimum (3.6) and
(n)
γN := Trn+1→N [|ΨN ihΨN |]
be its n-th reduced density matrix. There exists a unique probability measure µ on MH ,
the set of minimizers of EH (modulo a phase), such that, along a subsequence and for all
n∈N
Z
(n)
lim γ
N →∞ N
=
MH
dµ(u) |u⊗n ihu⊗n |
(3.19)
strongly in the S1 (Hn ) norm. In particular, if EH has a unique minimizer (modulo a
constant phase), then for the whole sequence
(n)
⊗n
lim γN = |u⊗n
H ihuH |.
(3.20)
N →∞
The ideas of the proof are essentially contained in [70, 154, 159], applied to a somewhat
different context however. We follow some clarifications given in [113, Section 3].
Proof. We have to prove the lower bound corresponding to (3.9). We start by writing
1
1
E(N )
(1)
(2)
=
hΨN , HN ΨN iHN = TrH [T γN ] + TrH2s [wγN ]
N
N
2
i
h
1
(2)
= TrH2s (T1 + T2 + w) γN
2
(3.21)
(1)
(2)
and we now have to describe the limits of the reduced density matrices γN and γN .
(n)
Since the sequences (γN )N ∈N are by definition bounded in S1 , by a diagonal extraction
argument we may assume that for all n ∈ N
(n)
γN ⇀∗ γ (n) ∈ S1 (Hns )
weakly-∗ in S1 (Hn ). That is, for every compact operator Kn over Hn we have
i
i
h
h
(n)
TrHn γN Kn → TrHn γ (n) Kn .
We are going to show that the limit is actually strong. For this it is sufficient (see [53, 164]
or [183, Addendum H]) to show that
i
i
h
h
(n)
(3.22)
TrHns γ (n) = TrHns γN = 1,
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
49
i.e. no mass is lost in the limit. We start by noting that, using Assumption (3.3),
i
h
1
(2)
eH ≥ TrH2s (T1 + T2 + w) γN
2
i
1 h
(2)
(2)
≥ (1 − β− ) Tr (T1 + T2 )γN − C Tr[γN ]
2 h
i
(1)
= (1 − β− ) Tr T γN
− C.
(1)
Thus TrH [T γN ] is uniformly bounded and, up to a possible further extraction, we have
(1)
(T + C0 )1/2 γN (T + C0 )1/2 ⇀∗ (T + C0 )1/2 γ (1) (T + C0 )1/2
for a certain constant C0 . Consequently
i
h
(1)
(1)
1 = TrH [γN ] = TrH (T + C0 )−1 (T + C0 )1/2 γN (T + C0 )1/2
i
h
→ TrH (T + C0 )−1 (T + C0 )1/2 γ (1) (T + C0 )1/2 = TrH [γ (1) ]
since (T + C0 )−1 is by Assumption (3.13) a compact operator. One obtains (3.22) similarly
by noting that


n
X
1
(n)
(1)
Tj γN 
TrH [T γN ] = TrHn 
n
j=1
Pn
is uniformly bounded in N and that j=1 Tj also has compact resolvent which allows for
a similar argument.
We thus have, for all n ∈ N
(n)
γN → γ (n)
strongly in trace-class norm, and in particular, for all bounded operator Bn on Hn
(n)
TrHn [γN Bn ] → TrHn [γ (n) Bn ].
Testing this convergence with Bn+1 = Bn ⊗ 1 we deduce
Trn+1 [γ (n+1) ] = γ (n)
and thus the sequence (γ (n) )n∈N describes a bosonic state with infinitely many particles
in the sense of Definition 3.3. We apply Theorem 3.5, which yields a measure µ ∈ P(SH).
In view of Assumption (3.3), the operator T1 + T2 + w is bounded below on H2 , say by
2CT . Since TrH2 γ (2) = 1 we may write
i
i
h
h
1
1
(2)
(2)
lim inf TrH2 (T1 + T2 + w) γN = lim inf TrH2 (T1 + T2 + w − 2CT ) γN + CT
N →∞ 2
N →∞ 2
i
h
1
≥ TrH2 (T1 + T2 + w − 2CT ) γ (2) + CT
2
i
h
1
= TrH2 (T1 + T2 + w) γ (2)
2
using Fatou’s lemma for positive operators. Using the linearity of the energy as a function
of γ (2) and (3.18)
Z
Z
1
E(N )
⊗2
⊗2
EH [u]dµ(u) ≥ eH ,
lim inf
≥
TrH2 (T1 + T2 + w) |u ihu | dµ(u) =
N →∞
N
u∈SH 2
u∈SH
50
NICOLAS ROUGERIE
which is the desired lower bound. The other statements of the theorem follow by noting
that equality must hold in all the previous inequalities and thus that µ may charge only
minimizers of Hartree’s functional.
It is clear from the preceding proof that the structure of bosonic states with infinitely
many particles plays the key role. The Hamiltonian itself could be chosen in a very abstract
way provided it includes a confining mechanism allowing to obtain strong limits. Several
examples are mentioned in [113, Section 3].
3.3. Systems with no bound states and the weak de Finetti theorem.
In the previous section we have used in a strong way the assumption that the system
was confined in the sense of (3.11)-(3.12). These assumptions are sufficient to understand
many physical cases, but it is highly desirable to be able to relax them. One is then
lead to study cases where the convergence of reduced density matrices is not better than
weak-∗, and to describe as exhaustively as possible the possible scenarii, in the spirit of
the concentration-compactness principle. A first step, before asking the question of how
compactness may be lost, consists in describing the weak limits themselves. It turns out
that we still have a very satisfying description. In fact, one could not have hoped for
better than the following theorem, proven in [113]:
Theorem 3.7 (Weak quantum de Finetti).
Let H be a complex separable Hilbert space and (ΓN )N ∈N a sequence of bosonic states with
ΓN ∈ S1 (HN
s ). We assume that for all n ∈ N
(n)
ΓN ⇀∗ γ (n)
(3.23)
in S1 (Hns ). Then there exists a unique probability measure µ ∈ P(BH) on the unit ball
BH = {u ∈ H, kuk ≤ 1} of H, invariant under the action of S 1 , such that
Z
(n)
|u⊗n ihu⊗n | dµ(u)
(3.24)
γ =
BH
for all n ≥ 0.
Remark 3.8 (On the weak quantum de Finetti theorem).
(1) Assumption (3.23) can always be satisfied in practice. Modulo a diagonal extraction, one may always assume that convergence holds along a subsequence. This
theorem thus exactly describes all the possible weak limits for a sequence of bosonic
N -body states when N → ∞.
(2) The fact that the measure lives over the unit ball in (3.24) is not surprising since
there may be a loss of mass in cases covered by the theorem. In particular it is
possible that γ (n) = 0 for all n and then µ = δ0 , the Dirac mass at the origin.
(3) The term weak refers to “weak convergence” and does not indicate that this result
is less general than the strong de Finetti theorem. It is in fact more general. To
see this, think of the case where no mass is lost, TrHn [γ (n) ] = 1. The measure µ
must then be supported on the sphere and convergence must hold in trace-class
norm. It is by the way sufficient to assume that TrHn [γ (n) ] = 1 for a certain n ∈ N,
and the convergence is strong for all n since the measure µ does not depend on n.
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
51
(4) Ammari and Nier have slightly more general results [4, 5, 6, 7]. In particular, it is
not necessary to start from a state with a fixed particle number. One can consider
a state on Fock space provided suitable bounds on its particle number (seen as a
random variable in this framework) are available.
(5) Uniqueness of the measure follows from a simple argument. Here we will mostly
be interested in the existence, which is sufficient for static problems. For timedependent problems on the contrary, uniqueness is crucial [5, 6, 7, 36].
Coming back to the derivation of Hartree’s theory, let us recall that the energy upper
bound (3.9) is always true. We are thus only looking for lower bounds. A case where
knowning the weak-∗ limit of reduced density matrices is then that of a weakly-∗ lower
semi-continuous functional. This remark may seem of marginal interest, but this case does
cover a number of physically relevant systems, those with no bound states.
We are going to prove the validity of Hartree’s approximation in this case, using Theorem 3.7. We shall work in Rd and assume that the potential V in (3.2) is non-trapping in
every direction:
V ∈ Lp (Rd ) + L∞ (Rd ), max 1, d/2 ≤ p < ∞, V (x) → 0 when |x| → ∞.
(3.25)
The assumption about the absence of bound states concerns the interaction potential w.
It is materialized par the inequality12
w
(3.26)
−∆+ ≥0
2
as operators. This means that w is not attractive enough for particles to form bound
states such as molecules etc ... Indeed, because of Assumption (3.4), −∆ + w2 can have
only negative eigenvalues, its essential spectrum starting at 0. In view of (3.26), it can in
fact not have any eigenvalue at all, and thus no eigenfunctions which are by definition the
bound states of the potential. A particular example is that of a purely repulsive potential
w ≥ 0.
Under Assumption (3.26), particles that might escape to infinity no longer see the onebody potential V and then necessarily carry a positive energy that can be neglected for a
lower bound to the total energy. Particles staying confined by the potential V are described
by the weak-∗ limits of density matrices. We can apply the weak de Finetti theorem to
the latter, and this leads to the next theorem, for whose statement we need the notation
eH (λ) :=
inf EH [u],
kuk2 =λ
0≤λ≤1
(3.27)
for the Hartree energy in the case of a loss of mass. Under assumption (3.26) it is not
difficult to show that for all 0 ≤ λ ≤ 1
eH (λ) ≥ eH (1) = eH
by constructing trial states made of two well-separated pieces of mass.
12Or an appropriate variant when one considers a different kinetic energy, cf Remark 3.2.
(3.28)
52
NICOLAS ROUGERIE
Theorem 3.9 (Derivation of Hartree’s theory in the absence of bound states).
Under the previous assumptions, in particular (3.25) and (3.26) we have
E(N )
= eH .
N →∞ N
be a sequence of approximate ground states, satisfying
lim
Let ΨN
hΨN , HN ΨN i = E(N ) + o(N )
in the limit N → ∞ and
(n)
γN := Trn+1→N [|ΨN ihΨN |]
be the corresponding n-th reduced density matrices. There exists a probability measure µ
supported on
o
n
MH = u ∈ BH, EH [u] = eH kuk2 = eH (1)
such that, along a subsequence and for all n ∈ N
Z
(n)
dµ(u) |u⊗n ihu⊗n |
γN ⇀ ∗
(3.29)
MH
in S1 (Hn ). In particular, if eH has a unique minimizer uH , with kuH k = 1, then for the
whole sequence,
strongly in trace-class norm.
(n)
lim γ
N →∞ N
⊗n
= |u⊗n
H ihuH |
(3.30)
Remark 3.10 (Case where mass is lost).
Note that it is possible for the convergence in the above statement to be only weak-∗,
which covers a certain physical reality. If the one-body potential is not attractive enough
to retain all particles, we will typically have a scenario where
(
eH (λ) = eH (1) for λc ≤ λ ≤ 1
eH (λ) < eH (1) for 0 ≤ λ < λc
where λc is a critical mass that can be bound by the potential V . In this case, eH (λ) will
not be achieved if λc < λ ≤ 1 and one will have a minimizer for Hartree’s energy only
for a mass 0 ≤ λ ≤ λc . If for example the minimizer uH at mass λc is unique modulo a
constant phase, Theorem 3.9 shows that
(n)
⊗n
γN ⇀∗ |u⊗n
H ihuH |
and one should note that the limit has a mass λnc < 1. This scenario actually happens in
the case of a “bosonic atom”, see Section 4.2 in [113].
Theorem 3.9 was proved in [113]. In order to be able to apply Theorem 3.7 we start
with the following observation:
Lemma 3.11 (Lower semi-continuity of an energy with no bound state).
(1) (2)
Under the previous assumptions, let γN , γN ≥ 0 be two sequences satisfying
(2)
TrH2 γN = 1,
(1)
(2)
γN = Tr2 γN
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
53
(k)
as well as γN ⇀∗ γ (k) weakly-∗ in S1 (Hk ) for k = 1, 2. Then
lim inf
N →∞
(1)
TrH [T γN ]
1
1
(2)
+ TrH2 [wγN ] ≥ TrH [T γ (1) ] + TrH2 [wγ (2) ].
2
2
(3.31)
Proof. We shall need two trunctation functions 0 ≤ χR , ηR ≤ 1 satisfying
2
χ2R + ηR
≡ 1, supp(χR ) ⊂ B(0, 2R), supp(ηR ) ⊂ B(0, R)c , |∇χR | + |∇ηR | ≤ CR−1 .
It is easy to show the IMS formula:
− ∆ = χR (−∆)χR + ηR (−∆)ηR − |∇χR |2 − |∇ηR |2
≥ χR (−∆)χR + ηR (−∆)ηR − CR−2 (3.32)
as an operator. For the one-body part of the energy we then easily have
(1)
(1)
(1)
TrH [T γN ] ≥ TrH [T χR γN χR ] + TrH [−∆ηR γN ηR ] + r1 (N, R)
with
(1)
2
V γN ]
r1 (N, R) ≥ −CR−2 + TrH [ηR
and thus
lim inf lim inf r1 (N, R) = 0
R→∞
N →∞
because in view of (3.25) we have
Z
(1)
(1)
2
2
ηR
(x)V (x)ρN (x)dx → 0 when R → ∞
TrH [ηR V γN ] =
(3.33)
Rd
(1)
(1)
uniformly in N . Here we have denoted ρN the one-body density of γN , formally defined
by
(1)
(1)
ρN (x) = γN (x, x),
(1)
where we identify γN and its kernel. The details of the proof of (3.33) are left to the
(1)
(1)
reader. One should use the a priori bound Tr[T γN ] ≤ C to obtain a bound on ρN in
some appropriate Lq space via Sobolev embeddings, then use Hölder’s inequality and the
fact that, for the conjugate exponent p,
2 ηR V p d → 0
L (R )
by assumption.
To deal with the interaction term we introduce
(2)
(2)
ρN (x, y) := γN (x, y; x, y)
54
NICOLAS ROUGERIE
(2)
(2)
the two-body density of γN (identified with its kernel γN (x′ , y ′ ; x, y)) and write
ZZ
(2)
(2)
w(x − y)ρN (x, y)dxdy
TrH2 [wγN ] =
d
d
Z ZR ×R
(2)
w(x − y)χ2R (x)ρN (x, y)χ2R (y)dxdy
=
Rd ×Rd
ZZ
(2)
2
2
(y)dxdy
w(x − y)ηR
(x)ρN (x, y)ηR
+
d
d
R ×R
ZZ
(2)
2
(y)dxdy
w(x − y)χ2R (x)ρN (x, y)ηR
+2
Rd ×Rd
ZZ
(2)
w(x − y)χ2R (x)ρN (x, y)χ2R (y)dxdy
=
d
d
Z Z R ×R
(2)
2
2
(y)dxdy
(x)ρN (x, y)ηR
w(x − y)ηR
+
Rd ×Rd
+ r2 (N, R)
(2)
(2)
= TrH2 [w χR ⊗ χR γN χR ⊗ χR ] + TrH2 [w ηR ⊗ ηR γN ηR ⊗ ηR ]
+ r2 (N, R)
where
lim inf lim inf r2 (N, R) = 0
R→∞
N →∞
by a standard concentration-compactness argument that we now sketch. One has to show
that
ZZ
(2)
2
(y)dxdy.
|w(x − y)|χ2R (x)ρN (x, y)ηR
lim lim
R→∞ N →∞
Rd ×Rd
2 (y) ≤ 1
Since χ2R (x)|w(x − y)|η4R
|x−y|≥R |w(x − y)|, the term
ZZ
(2)
2
(y)dxdy
|w(x − y)|χ2R (x)ρN (x, y)η4R
Rd ×Rd
can be dealt with similarly as (3.33). We leave again the details to the reader. There thus
remains to control
ZZ
(2)
2
2
(y) − η4R
(y) dxdy.
|w(x − y)|χ2R (x)ρN (x, y) ηR
Rd ×Rd
for which we claim that
Z Z
(2)
2
2
lim lim
|w(x − y)|χ2R (x)[ηR
(y) − η4R
(y)]ρNk (x, y) dx dy
R→∞ k→∞
Z Z
(2)
|w(x − y)|1(R ≤ |y| ≤ 8R)ρNk (x, y) dx dy = 0
≤ lim lim
R→∞ k→∞
for an appropriate subsequence (Nk ). Let us introduce the concentration functions
ZZ
(2)
QN (R) :=
|w(x − y)|1(|y| ≥ R)ρN (x, y) dx dy.
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
55
For every N , the function R 7→ QN (R) is decreasing on [0, ∞). Moreover,
(2)
0 ≤ QN (R) ≤ TrH2 [|w|γN ] ≤ C0
i
h
(1)
using our assumptions on w and the a priori bound Tr T γN ≤ C again. Therefore,
by Helly’s selection principle, there exists a subsequence Nk and a decreasing function
Q : [0, ∞) → [0, C0 ] such that QNk (R) → Q(R) for all R ∈ [0, ∞). Since limR→∞ Q(R)
exists, we conclude that
lim lim (QNk (R) − QNk (8R)) = lim (Q(R) − Q(8R)) = 0.
R→∞ k→∞
R→∞
This is the desired convergence.
At this stage we thus have
1
(2)
TrH2 [wγN ] ≥
2
(1)
lim inf TrH [T γN ] +
N →∞
1
(2)
Tr 2 [w χR ⊗ χR γN χR ⊗ χR ]
2 H
1
(1)
(2)
+ lim inf lim inf TrH [−∆ηR γN ηR ] + TrH2 [w ηR ⊗ ηR γN ηR ⊗ ηR ] (3.34)
R→∞ N →∞
2
The terms on the second line give the right-hand side of (3.31). Indeed, recalling that
T = −∆ + V , using Fatou’s lemma for −∆ ≥ 0 and the fact that
(1)
lim inf lim inf TrH [T χR γN χR ] +
R→∞
N →∞
(1)
χR γN χR → χR γ (1) χR
in norm since χR has compact support, we have
(1)
lim inf TrH [T χR γN χR ] ≥ TrH [T χR γ (1) χR ].
N →∞
It then suffices to recall that χR → 1 pointwise to conclude. The interaction term is dealt
with in a similar way, using the strong convergence
(2)
χR ⊗ χR γN χR ⊗ χR → χR ⊗ χR γ (2) χR ⊗ χR
and then the pointwise convergence χR → 1.
The terms on the third line of (3.34) form a positive contribution that we may drop
from the lower bound. To see this, note that since 0 ≤ ηR ≤ 1 we have ηR ⊗ 1 ≥ ηR ⊗ ηR
and (with Tr2 the partial trace with respect to the second variable)
(1)
(2)
ηR γN ηR ≥ Tr2 [ηR ⊗ ηR γN ηR ⊗ ηR ],
(2)
which gives, by symmetry of γN ,
1
(2)
TrH2 [w ηR ⊗ ηR γN ηR ⊗ ηR ]
2
i
h
1
(2)
≥ TrH2 ((−∆) ⊗ 1 + 1 ⊗ (−∆) + w) ηR ⊗ ηR γN ηR ⊗ ηR
2
13
and it is not difficult to see that Assumption (3.26) implies
(1)
TrH [−∆ηR γN ηR ] +
(−∆) ⊗ 1 + 1 ⊗ (−∆) + w ≥ 0
13Just decouple the center of mass from the relative motion of the two particles, i.e. make the change
of variables (x1 , x2 ) 7→ (x1 + x2 , x1 − x2 ).
56
NICOLAS ROUGERIE
which ensures the positivity of the third line of (3.34) and concludes the proof.
We now conclude the
Proof of Theorem 3.9. Starting from a sequence ΓN = |ΨN ihΨN | of N -body states we
extract subsequences as in the proof of Theorem 3.6 to obtain
(n)
Using Lemma 3.11, we obtain
ΓN ⇀∗ γ (n) .
1
(2)
(1)
lim inf N TrHN [HN ΓN ] = lim inf TrH [T ΓN ] + TrH2 [wΓN ]
N →∞
N →∞
2
1
≥ TrH [T γ (1) ] + TrH2 [wγ (2) ]
2
and there remains to apply Theorem 3.7 to the sequence (γ (n) )n∈N to obtain
Z
−1
EH [u]dµ(u) ≥ eH
lim inf N TrHN [HN ΓN ] ≥
N →∞
BH
R
using (3.28) and the fact that BH dµ(u) = 1. Once again, the other conclusions of the
theorem easily follow by inspecting the cases of equality in the previous estimates.
−1
For later use, we note that during the proof of Lemma 3.11 we have obtained the
intermediary result (3.34) without using the assumption that w has no bound state. We
write this as a lemma that we shall use again in Chapter 6.
Lemma 3.12 (Energy localization).
Under assumptions (3.4) and (3.25), let ΨN be a sequence of almost minimizers for E(N ):
and
(k)
γN
hΨN , HN ΨN i = E(N ) + o(N )
the associated reduced density matrices. We have
1
E(N )
(2)
(1)
= lim inf TrH [T γN ] + TrH2 [wγN ] ≥
N →∞
N →∞
N
2
1
(1)
(2) ⊗2
lim inf lim inf TrH [T χR γN χR ] + TrH2 [w χ⊗2
R γN χR ]
R→∞ N →∞
2
1
(1)
⊗2 (2) ⊗2
+ lim inf lim inf TrH [−∆ηR γN ηR ] + TrH2 [w ηR
γN ηR ] (3.35)
R→∞ N →∞
2
q
where 0 ≤ χR ≤ 1 is C 1 , with support in B(0, R), and ηR = 1 − χ2R .
lim inf
3.4. Links between various structure theorems for bosonic states.
We have just introduced two structure theorems for many-particles bosonic systems.
These indicate that, morally, if ΓN is a N -body bosonic state on a separable Hilbert
space H, there exists a probability measure µ ∈ P(H) on the one-body Hilbert space such
that
Z
(n)
|u⊗n ihu⊗n |dµ(u)
(3.36)
ΓN ≈
u∈H
when N is large and n is fixed. Chapters 4 and 5 of these notes are for a large devoted to
the proofs of these theorems “à la de Finetti”. As mentioned in Remark 3.8, Theorem 3.7
is actually more general than Theorem 3.5, and we will thus prove the former.
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
57
In order to better appreciate the importance of the weak theorem in infinite dimensional
spaces, we emphasize that the key property allowing to prove the strong theorem is the
(n)
consistency (3.15). When starting from the reduced density matrices ΓN of a N -body
state ΓN and extracting weakly-∗ convergent subsequences to define a hierarchy γ (n) n∈N ,
(n)
ΓN ⇀∗ γ (n) ,
one only has
i
h
i
h
(n)
(n+1)
= lim γN = γ (n) .
Trn+1 γ (n+1) ≤ lim Trn+1 γN
N →∞
because the trace is not continuous
Obviously, the relation
14
N →∞
for the weak-∗ topology, only lower semi-continuous.
i
h
Trn+1 γ (n+1) ≤ γ (n)
(3.37)
is not sufficient to prove a de Finetti theorem. A simple counter-example is given by the
sequence of reduced density matrices of a one-body state v ∈ SH:
γ (0) = 1, γ (1) = |vihv|, γ (n) = 0 for n ≥ 2.
In these notes we have chosen to prove the weak de Finetti theorem as constructively
as possible. Before we announce the plan of the proof, we shall say a few words of the
much more abstract approach of the historical references [192, 94]. These papers actually
contain a version of the theorem that is even stronger than Theorem 3.7. This result
applies to “abstract” states that are not necessarily normal or bosonic. These may be
defined as follows:
Definition 3.13 (Abstract state with infinitely many particles).
Nn
H be the
Let H be a complex separable Hilbert space and for all n ∈ N, Hn =
corresponding n-body space. We call an abstract state with infinitely many particles a
sequence (ω (n) )n∈N where
• ω (n) is an abstract n-body state: ω (n) ∈ (B(Hn ))∗ , the dual of the space of bounded
operators on Hn , ω (n) ≥ 0 and
ω (n) (1Hn ) = 1.
• ω (n) is symmetric in the sense that
ω (n) (B1 ⊗ . . . ⊗ Bn ) = ω (n) Bσ(1) ⊗ . . . ⊗ Bσ(n)
for all B1 , . . . , Bn ∈ B(H) and all permutation σ ∈ Σn .
• the sequence (ω (n) )n∈N is consistent:
(3.38)
ω (n+1) (B1 ⊗ . . . ⊗ Bn ⊗ 1H ) = ω (n) (B1 ⊗ . . . ⊗ Bn )
for all B1 , . . . , Bn ∈ B(H).
(3.39)
(3.40)
An abstract state is in general not normal, i.e. it does not fit in the following definition:
14This is a fancy way of characterizing infinite dimensional spaces.
58
NICOLAS ROUGERIE
Definition 3.14 (Normal state, locally normal state).
Let H be a complex separable Hilbert space and (ω (n) )n∈N be an abstract state with
infinitely many particles. We say that (ω (n) )n∈N is locally normal if ω (n) is normal for all
n ∈ N, i.e. there exists γ (n) ∈ S1 (Hn ) a trace-class operator such that
ω (n) (Bn ) = TrHn [γ (n) Bn ]
for all Bn ∈ B(Hn ).
(3.41)
Identifying trace-class operators with the associated normal states we readily see that
Definition 3.3 is a particular case of abstract state with infinitely many particles. Note
that (by the spectral theorem), the set of convex combinations of pure states (orthogonal
projectors) coı̈ncides with the trace class. A non-normal abstract state is thus not a mixed
state, i.e. not a statistical superposition of pure states. The physical interpretation of a
non-normal state is thus not obvious, at least in the type of settings that these notes are
concerned with.
The consistency notion (3.40) is the natural generalization of (3.15) but it is important
to note that the symmetry (3.39) is weaker than bosonic symmetry. It in fact corresponds
to classicaly indistinguishable particles (the modulus of the wave-function is symmetric
but not the wave-function itself). One may for example note that if ω (n) is normal in the
sense of (3.41), then γ (n) ∈ S1 (Hn ) satisfies
Uσ γ (n) Uσ∗ = γ (n)
where Uσ is the unitary operator permuting the n particles according to σ ∈ Σn . Bosonic
symmetry corresponds to the stronger constraint
Uσ γ (n) = γ (n) Uσ∗ = γ (n) ,
cf Section 1.2.
We also have a notion of product state that generalizes Definition 3.4.
Definition 3.15 (Abstract product state with infinitely many particles).
We call abstract product state an abstract state with infinitely many particles such that
ω (n) = ω ⊗n
for all n ∈ N, where ω ∈
ω(1H ) = 1).
(B(H))∗
(3.42)
is an abstract one-body state (in particular ω ≥ 0 and
The most general form of the quantum de Finetti theorem15 says that every abstract
state with infinitely many particles is a convex combination of product states:
Theorem 3.16 (Abstract quantum de Finetti).
Let H be a complex separable Hilbert space and (ω (n) )n∈N an abstract bosonic state
with infinitely many particles built on H. There exists a unique probability measure
µ ∈ P ((B(H))∗ ) on the dual of the space of bounded perators on H such that
and
µ ({ω ∈ (B(H))∗ , ω ≥ 0, ω(1H ) = 1}) = 1
ω (n) =
Z
ω ⊗n dµ(ω)
15At least the most general form the author is aware of.
(3.43)
(3.44)
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
59
for all n ≥ 0.
Remark 3.17 (On the abstract quantum de Finetti theorem).
(1) This result was first proven by Størmer [192]. Hudson and Moody [94] then gave a
simpler proof by adapting the Hewitt-Savage proof of the classical de Finetti theorem 2.1: they prove that product states are the extremal points of the convex set of
abstract states with infinitely many particles. The existence of the measure is then
a consequence of the Choquet-Krein-Milman theorem. This approach does require
the notion of abstract states and does not give a direct proof of Theorem 3.5.
(2) Hudson and Moody [94] deduce the strong de Finetti theorem from the abstract
theorem. An adaptation of their method (see [113, Appendix A]) shows that the
weak de Finetti theorem is also a consequence of the abstract theorem.
(3) This theorem has been used to derive Hartree-type theories for abstract states
without bosonic symmetry in [70, 159, 154]. To recover the usual Hartree theory
one must be able to show that the limit state is (locally) normal. In finite dimensional spaces, B(H) of course coı̈ncides with the space of compact operators, which
implies that any abstract state is normal. This difficulty thus does not occur in
this setting.
(4) There also exist quantum generalizations of the probabilistic versions of the classical de Finetti theorem (that is, those dealing with sequences of random variables [2, 96]), see e.g. [107]. These are formulated in the context of free probability
and require additional symmetry assumptions besides permutation symmetry.
At this stage we thus have the scheme (“deF” stands for de Finetti)
abstract deF ⇒ weak deF ⇒ strong deF ,
(3.45)
but the proof of the weak de Finetti theorem we are going to present follows a different
route, used in [113, 116]. It starts from the finite dimensional theorem16:
finite-dimensional deF ⇒ weak deF ⇒ strong deF .
(3.46)
This approach leads to a somewhat longer proof than the scheme (3.45) starting from the
Hudson-Moody proof of Theorem 3.16. This detour is motivated by five main practical
and aesthetic reasons:
(1) The proof following (3.46) is simpler from a conceptual point of view: it requires
neither the notion of abstract states nor the use of the Choquet-Krein-Milman
theorem.
(2) Thanks to recent progress, due mainly to the quantum information community [39,
38, 90, 106, 71, 116], we have a completely constructive proof the finite dimensional
quantum de Finetti theorem at our disposal. One first proves explicit estimates
thanks for a construction for finite N , in the spirit of the Diaconis-Freedman
approach to the classical case. Then one passes to the limit as in the proof of the
Hewitt-Savage theorem we have presented in Section 2.2.
16In finite dimension there is no need to distinguish between the weak and the strong version.
60
NICOLAS ROUGERIE
(3) The first implication in Scheme (3.46) is also essentially constructive, thanks to
Fock-space localization techniques used e.g. in [3, 55, 112]. These tools are inherited from the so-called “geometric” methods [66, 67, 180, 182] that adapt to the
N -body problem localization ideas natural in the one-body setting. These allow
(amongst other things) a fine description of the lack of compactness due to loss of
mass at infinity, in the spirit of the concentration-compactness principle [136, 137].
(4) In particular, the proof of the first implication in (3.46) yields a few corollaries which will allow us to prove the validity of Hartree’s theory in the general
case. When the assumptions made in Section 3.3 do not hold, the weak de Finetti
theorem and its proof according to (3.45) are not sufficient to conclude: Particles escaping to infinity may form negative-energy bound states. The localization
methods we are going to discuss will allow us to analyze this phenomenon.
(5) In Chapter 7 we will deal with a case where the interaction potential depends on
N to derive non-linear Schrödinger theories in the limit. This amounts to taking
a limit where w converges to a Dirac mass simultaneously to the N → ∞ limit. In
this case, compactness arguments will not be sufficient and the explicit estimates
we shall obtain along the proof of the finite dimensional de Finetti theorem will
come in handy.
An alternative point of view on the proof strategy (3.46) is given by the Ammari-Nier
approach [4, 5, 6, 7], based on semi-classical analysis methods. The relation between the
two approaches will be discussed below.
As noted above, the second implication in (3.46) is relatively easy. The following chapters deal with the first two steps of the strategy. They contain the proof of the weak de
Finetti theorem and several corollaries and intermediary results. The finite dimensional
setting (where the distinction between the strong and the weak theorems is irrelevant) is
discussed in Chapter 4. The localization methods allowing to prove the first implication
in (3.46) are the subject of Chapter 5.
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
61
4. The quantum de Finetti theorem in finite dimensonal spaces
This chapter deals with the starting point of the proof strategy (3.46), that is a proof of
the strong de Finetti theorem in the case of a finite dimensional complex Hilbert space H,
dim H = d.
In this case, strong and weak-∗ convergences in S1 (Hn ) are the same and thus there is
no need to distinguish between the strong and the weak de Finetti theorem. The main
advantage of working in finite dimensions is the possibility to obtain explicit estimates, in
the spirit of the Diaconis-Freedman theorem (with a completely different method, though).
We are going to prove the following result, which gives bounds in trace-class norm:
Theorem 4.1 (Quantitative quantum de Finetti).
(n)
N
Let ΓN ∈ S1 (HN
s ) be a bosonic state over Hs and γN its reduced density matrices. There
exists a probability measure µN ∈ P(SH) such that, denoting
Z
|u⊗N ihu⊗N |dµN (u)
(4.1)
Γ̃N =
u∈SH
(n)
the associated state and γ̃N its reduced density matrices, we have
2n(d + 2n)
(n)
(n) eN ≤
TrHn γN − γ
N
for all n = 1 . . . N .
(4.2)
Remark 4.2 (On the finite dimensional quantum de Finetti theorem).
(1) This result is due to Christandl, König, Mitchison and Renner [39], important
earlier work being found in [106] and [71]. One may find developments along
theses lines in [38, 90, 116]. The quantum information community also considered
several variants, see for example [33, 39, 40, 41, 162, 27].
(2) One can add a step in the strategy (3.46):
quantitative deF ⇒ finite dimensional deF ⇒ weak deF ⇒ strong deF .
(4.3)
Indeed, in finite dimension one may identify the sphere SH with a usual, compact,
Euclidean sphere (with dimension 2d − 1, i.e. the unit sphere in R2d ). The space of
probability measures on SH is then compact for the usual weak topology and one
may extract from µN a converging subsequence to prove Theorem 3.5 in the case
where dim H < ∞, exactly as we did to deduce Theorem 2.1 from Theorem 2.2 in
Section 2.2.
(3) The bound (4.2) is not optimal. One may in fact obtain the estimate
2nd
(n)
(n) ,
(4.4)
eN ≤
TrHn γN − γ
N
with the same construction, see [38, 39, 116]. The proof we shall present only
gives (4.2) but seems more instructive to me. For the applications we have in
mind, n will always be fixed anyway (equal to 2 most of the time), and in this
case (4.2) and (4.4) give the same order of magnitude in terms of N and d.
62
NICOLAS ROUGERIE
(4) The bound (4.4) in the quantum case is the equivalent of the estimate in dn/N
mentioned in Remark 2.4 for the classical case. One may ask if this order of
magnitude is optimal. It clearly is with the construction we are going to use, but
it would be very interesting to know if one can do better with another construction.
In particular, can one find a bound independent from d, reminiscent of (2.14) in
the Diaconis-Freedman theorem ?
The construction of Γ̃N is taken from [39]. It is particularly simple but it does use
in a strong manner the fact that the underlying Hilbert space has finite dimension. The
approach we shall follow for the proof of Theorem 4.1 is originally due to Chiribella [38].
We are going to prove an explicit formula giving the density of Γ̃N as a function of those of
ΓN , in the spirit of Remark 2.3. This formula implies (4.2) in the same manner as (2.19)
implies (2.14).
In Section 4.1 we present the construction, state Chiribella’s explicit formula and obtain
Theorem 4.1 as a corollary. Before giving a proof of Chiribella’s result, it is useful to
discuss some informal motivation and some heuristics on the Christandl-König-MitchisonRenner (CKMR) construction, which happens to be connected to well-known ideas of
semi-classical analysis. This is the purpose of Section 4.2. Finally, we prove Chiribella’s
formula in Section 4.3, following the approach of [116]. It has been independently found
by Lieb and Solovej [129] (with a different motivation), and was inspired by the works
of Ammari and Nier [5, 6, 7]. Related considerations appeared also in [104, Chapter 3].
Other proofs are available in the literature, cf [38] and [90].
4.1. The CKMR construction and Chiribella’s formula.
We first note that the Diaconis-Freedman construction introduced before is purely classical since it is based on the notion of empirical measure, which has no quantum counterpart.
A different approach is thus clearly necessary for the proof of Theorem 4.1.
In a finite dimensional space, one may identify the unit sphere SH = {u ∈ H, kuk = 1}
with a Euclidean sphere. One may thus equip it with a uniform measure (Haar measure of
the rotation group, simply the Lebesgue measure on the Euclidean sphere), that we shall
denote du, taking the convention that
Z
du = 1.
SH
We then have a nice resolution of the identity as a simple consequence of the invariance
of du under rotations (see Section 4.3 below for a proof). We state this as a lemma:
Lemma 4.3 (Schur’s formula).
Let H be a complex finite dimensional Hilbert space and HN the corresponding bosonic
N -body space. Then
Z
dim HN
s
SH
|u⊗N ihu⊗N | du = 1HN .
The idea of Christandl-König-Mitchison-Renner is to simply define
⊗N
⊗N
dµN (u) := dim HN
Γ
|u
ihu
|
du
N
s TrHN
s
⊗N
N
⊗N
= dim Hs TrHN
u , ΓN u
du,
s
(4.5)
(4.6)
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
i.e. to take
Γ̃N =
dim HN
s
Z
SH
63
|u⊗N ihu⊗N | u⊗N , ΓN u⊗N du.
(4.7)
Chriibella’s observation17 is the following:
Theorem 4.4 (Chiribella’s formula).
With the previous definitions, it holds
n N + n + d − 1 −1 X N (ℓ)
(n)
γ̃N =
γ ⊗s 1Hn−ℓ
n
ℓ N
(4.8)
ℓ=0
with the convention
(ℓ)
γN ⊗s 1Hn−ℓ =
X
1
ℓ
(γN
)σ(1),...,σ(ℓ) ⊗ (1Hn−ℓ )σ(ℓ+1),...,σ(n)
ℓ! (n − ℓ)!
σ∈Sn
ℓ )
where (γN
σ(1),...,σ(ℓ) acts on the σ(1) . . . , σ(ℓ) variables.
From this result we deduce a simple proof of the quantitative de Finetti theorem:
Proof of Theorem 4.1. We proceed as in (2.20). Only the first term in the sum (4.8) is
really relevant:
(n)
(n)
(n)
γ̃N − γN = (C(d, n, N ) − 1)γN + B = −A + B
where
(4.9)
N!
(N + d − 1)!
< 1,
(N + n + d − 1)! (N − n)!
and A, B are positive operators. We have
i
h
(n)
(n)
TrHn [−A + B] = Tr γ̃N − γN = 0,
C(d, n, N ) =
and thus, by the triangle inequality,
(n)
(n) Tr γ̃N − γN ≤ Tr A + Tr B = 2 Tr A = 2(1 − C(d, n, N )).
Next, the elementary inequality
C(d, n, N ) =
n−1
Y
j=0
N −j
≥
N +j+d
gives
which is the desired result.
1−
2n + d − 2
N +d+n−1
n
2n(d + 2n)
(n)
(n) Tr γN − γ̃N ≤
,
N
≥1−n
2n + d − 2
N +d+n−1
(4.10)
Everything now relies on the proof of Theorem 4.4, which is the subject of Section 4.3.
Before we give it, some heuristics regarding the relevance of the construction (4.7) shall
be discussed.
17In the quantum information vocabulary this is formulated as a relation between “optimal cloning”
and “optimal measure and prepare channels”.
64
NICOLAS ROUGERIE
4.2. Heuristics and motivation.
Schur’s formula (4.5) expresses the fact that the family u⊗N u∈SH forms an overcomplete basis of HN
s . Such a basis labeled by a continuous parameter is reminiscent of a
coherent state decomposition [103, 197]. This basis in fact turns our to be “less and less
over-complete” when N gets large. Indeed, we clearly have
(4.11)
hu⊗N , v ⊗N iHN = hu, viN
H → 0 when N → ∞
as soon as u and v are not exactly colinear. The basis u⊗N u∈SH thus becomes “almost
orthonormal” when N tends to infinity.
In the vocabulary of semi-classical analysis [120, 184, 16], trying to write
Z
dµN (u)|u⊗N ihu⊗N |
ΓN =
u∈SH
amounts to looking for an upper symbol µN representing ΓN . In fact, it so happens [184]
that one may always find such a symbol, only µN is in general not a positive measure. The
problem we face is to find a way to approximate the upper symbol (for which no explicit
expression as a function of the state itself exists, by the way) with a positive measure.
Note that if the coherent state basis were orthogonal, then the upper symbol would
be positive, simply by positivity of the state. Since we noted that the family u⊗N u∈SH
becomes “almost orthonormal” when N gets large, it is very natural to expect that the
upper symbol may be approximate by a positive measure in this limit.
On the other hand, the measure introduced in (4.6) is exactly what one calls the lower
symbol of the state ΓN . One of the reasons why lower and upper symbols were introduced
is that these two a priori different objects have a tendency to coı̈ncide in semi-classical
limits. But the N → ∞ limit we are concerned with may indeed be seen as a semi-classical
limit and it is thus very natural to take the lower symbol as an approximation of the upper
symbol in this limit.
One can motivate this choice in a slightly more precise way. Assume we have a sequence
of N -body states defined starting from an upper symbol independent of N ,
Z
N
µsup (u)|u⊗N ihu⊗N |du,
ΓN = dim(Hs )
u∈SH
and let us compute the corresponding lower symbols:
Z
2
⊗N
⊗N
N
inf
µsup (u) hu⊗N , v ⊗N i du
µN (v) = hv , ΓN v i = dim(Hs )
Zu∈SH
µsup (u) |hu, vi|2N du.
= dim(HN
s )
u∈SH
In view of the observation (4.11) and the necessary invariance of µsup under the action of
S 1 it is clear that we have
sup
µinf
(v) when N → ∞.
N (v) → µ
In other words, the lower symbol is, for large N , an approximation of the upper symbol, that has the advantage of being positive. Without consituting a rigorous proof of
Theorem 4.1, this point of view shows that the CKMR constuction is extremely natural.
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
65
4.3. Chiribella’s formula and anti-Wick quantization.
The proof of Theorem 4.4 we are going to present uses the second quantization formalism. We start with a very useful lemma. In the vocabulary alluded to in the previous
section it says that a state is entirely characterized by its lower symbol, a well-known
fact [184, 103].
Lemma 4.5 (The lower symbol determines the state).
If an operator γ (k) on Hks satisfies
hu⊗k , γ (k) u⊗k i = 0
then γ (k) ≡ 0.
for all u ∈ H,
(4.12)
Proof. We use the symmetric tensor product
X
1
Ψℓ (xσ(1) , ..., xσ(ℓ) )Ψk−ℓ (xσ(ℓ+1) , ..., xσ(k) )
Ψk ⊗s Ψℓ (x1 , ..., xk ) = p
ℓ!(k − ℓ)!k! σ∈S
k
Hℓ
Hk−ℓ
for two vectors Ψℓ ∈
and Ψk−ℓ ∈
and assume that (4.12) holds for all u ∈ H.
Picking two unit vectors u, v, replacing u by u + tv in (4.12) and taking the derivative
with respect to t (which must be zero), we obtain
E
D
ℜ u⊗k−1 ⊗s v, γ (k) u⊗k = 0.
Next, replacing u by u + itv and taking the derivative with respect to t we get
E
D
ℑ u⊗k−1 ⊗s v, γ (k) u⊗k = 0.
Hence for all u and v
D
E
u⊗k−1 ⊗s v, γ (k) u⊗k = 0.
(4.13)
Doing the same manipulations but taking now second derivatives with respect to t we also
have
E
D
E
D
u⊗k−1 ⊗s v, γ (k) u⊗k−1 ⊗s v + 2 u⊗k−2 ⊗s v ⊗2 , γ (k) u⊗k = 0.
But, on the other hand, replacing u by u + tv and u + itv in (4.13) and taking first
derivatives we get
E
D
E D
u⊗k−1 ⊗s v, γ (k) u⊗k−1 ⊗s v + u⊗k−2 ⊗s v ⊗2 , γ (k) u⊗k = 0.
Combining the last two equations we infer
D
E
u⊗k−1 ⊗s v, γ (k) u⊗k−1 ⊗s v = 0
for all unit vectors u and v. Taking v of the form v = v1 ± ve1 then v = v1 ± ie
v1 and
iterating the argument, we conclude
hv1 ⊗s v2 ⊗s . . . ⊗s vk , γ (k) ve1 ⊗s ve2 ⊗s . . . ⊗s vek i = 0
for all vj , vej ∈ H. Vectors of the form v1 ⊗s v2 ⊗s . . . ⊗s vk form a basis of Hks , thus the
proof is complete.
For self-containedness we use the previous lemma to give a short proof of Schur’s formula (4.5):
66
NICOLAS ROUGERIE
Proof of Lemma 4.3. In view of Lemma 4.5, it suffices to show that
Z
dim HN
|hu, vi|2N du = 1
s
SH
for all v ∈ SH. Pick v and ṽ in SH and U a unitary mapping such that U v = ṽ. Then, by
invariance of du under rotations,
Z
Z
Z
2N
2N
N
N
dim HN
|hu,
vi|
du
=
dim
H
|hU
u,
U
vi|
du
=
dim
H
|hu, ṽi|2N du
s
s
s
SH
SH
and thus the quantity
dim HN
s
Z
SH
SH
|hu, vi|2N du
does not depend on v. By Lemma 4.5 this implies that
Z
dim HN
|u⊗N ihu⊗N | du = c1HN
s
SH
for some constant c. Taking the trace of both sides of this equation shows that c = 1 and
the proof is complete.
In the sequel we use standard bosonic creation and annihilation operators. For all
f ∈ H, we define the creation operator a∗ (f ) : Hsk−1 → Hks by


X
X
fσ(1) ⊗ ... ⊗ fσ(k−1)  = (k)−1/2
a∗ (f ) 
fσ(1) ⊗ ... ⊗ fσ(k)
σ∈Sk−1
σ∈Sk
where in the right-hand side we set fk = f . The annihilation operator a(f ) : Hk+1 → Hk
is the formal adjoing of a∗ (f ) (whence the notation), defined by


X
X a(f ) 
fσ(1) ⊗ ... ⊗ fσ(k+1)  = (k + 1)1/2
f, fσ(1) fσ(2) ⊗ ... ⊗ fσ(k)
σ∈Sk+1
σ∈Sk+1
for all f, f1 , ..., fk in H. These operators satisfy the canonical commutation relations (CCR)
[a(f ), a(g)] = 0,
[a∗ (f ), a∗ (g)] = 0,
[a(f ), a∗ (g)] = hf, giH .
(4.14)
(n)
One of the uses of these objects is that the reduced density matrices γN of a bosonic
state ΓN are characterized by the relations18
(n)
hv ⊗n , γN v ⊗n i =
(N − n)!
[a∗ (v)n a(v)n ΓN ] .
TrHN
s
N!
(n)
(4.15)
Lemma 4.5 guarantees that this determines γN completely. The definition above is called
a Wick quantization: Creation and annihilation operators appear in the normal order, all
creators on the left and all annihilators on the right.
The key observation in the proof of Theorem 4.4 is that the density matrices of the
state (4.7) can be alternatively defined from ΓN via an anti-Wick quantization where
creation and annihilation operators appear in anti-normal order: All annihilators on the
left and all creators on the right.
18We recall our convention that Tr[γ (n) ] = 1.
N
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
67
Lemma 4.6 (The CKMR construction and anti-Wick quantization).
(n)
Let Γ̃N be defined by (4.7) and γ̃N be its reduced density matrices. We have
(n)
hv ⊗n , γ̃N v ⊗n i =
for all v ∈ H.
(N + d − 1)!
Tr N [a(v)n a∗ (v)n ΓN ]
(N + n + d − 1)! Hs
(4.16)
Proof. It suffices to consider the case of a pure state ΓN = |ΨN ihΨN | and write
Z
N
⊗k (k) ⊗k
du|hu⊗N , ΨN i|2 |hu⊗k , v ⊗k i|2
hv , γ
eN v i = dim Hs
ZSH
N
= dim Hs
du|hu⊗(N +k) , v ⊗k ⊗ ΨN i|2
SH
Z
N!
N
=
dim Hs
du|hu⊗(N +k) , a∗ (v)k ΨN i|2
(N + k)!
SH
dim HN
N!
s
ha(v)k ΨN , a(v)k ΨN i
(N + k)! dim HsN +k
(N + d − 1)!
hΨN , a(v)k a∗ (v)k ΨN i
=
(N + k + d − 1)!
=
using Schur’s lemma (4.5) in HsN +k in the third line, the fact that a∗ (v) is the adjoint of
a(v) in the fourth line and recalling that
N +d−1
N
dim Hs =
,
(4.17)
d−1
the number of ways of choosing N elements from d, allowing repetitions, without taking
the order into account. This is the number of orthogonal vectors of the form
ui1 ⊗s . . . ⊗s uiN ,
(i1 , . . . , iN ) ∈ {1, . . . , d}
one may form starting from an orthogonal basis (u1 , . . . , ud ) of H, and these form an
orthogonal basis of HN .
The way forward is now clear: we have to compare polynomials in a∗ (v) and a(v) written
in normal and anti-normal order. This standard operation leads to the final lemma of the
proof:
Lemma 4.7 (Normal and anti-normal order).
Let v ∈ SH. We have
n X
n n! ∗ k
n ∗
n
a(v) a (v) =
a (v) a(v)k for all n ∈ N.
k k!
(4.18)
k=0
Proof. The computation is made easier by recalling the expression for the n-th Laguerre
polynomial
n X
n (−1)k k
x .
Ln (x) =
k!
k
k=0
These polynomials satisfy the recurrence relation
(n + 1)Ln+1 (x) = (2n + 1)Ln (x) − xLn (x) − nLn−1 (x)
68
NICOLAS ROUGERIE
and one may see that (4.18) may be rewritten
n
X
a(v)n a∗ (v)n =
cn,k a∗ (v)k a(v)k
(4.19)
k=0
where the cn,k are the coefficients of the polynomial
L̃n (x) := n! Ln (−x).
It thus suffices to show that, for any n ≥ 1,
a(v)n+1 a∗ (v)n+1 = a∗ (v)a(v)n a∗ (v)n a(v) + (2n + 1)a(v)n a∗ (v)n − n2 a(v)n−1 a∗ (v)n−1 .
Note the order of creation and annihilation operators in the first term of the right-hand
side: knowing a normal-ordered representation of a(v)n a∗ (v)n and a(v)n−1 a∗ (v)n−1 we
deduce a normal-ordered representation of the left-hand side.
A repeated application of the CCR (4.14) gives the relations
a(v)a∗ (v)n = a∗ (v)n a(v) + na∗ (v)n−1
a(v)n a∗ (v) = a∗ (v)a(v)n + na(v)n−1
(4.20)
Then
a∗ (v)a(v)n a∗ (v)n a(v) = a(v)n a∗ (v)n+1 a(v) − na(v)n−1 a∗ (v)n a(v)
= a(v)n+1 a∗ (v)n+1 − (n + 1)a(v)n a∗ (v)n
and the proof is complete.
− na(v)n a∗ (v)n + n2 a(v)n−1 a∗ (v)n−1 ,
The final formula (4.8) is deduced by combining Lemmas 4.5, 4.6 and 4.7 with (4.15),
simply noting that for j ≤ n
N!
(j)
Tr a∗ (v)j a(v)j ΓN =
hv ⊗j , ΓN v ⊗j i
(N − j)!
N!
(j)
=
hv ⊗n , ΓN ⊗ 1⊗n−j v ⊗n i
(N − j)!
N!
(j)
=
hv ⊗n , ΓN ⊗s 1⊗n−j v ⊗n i.
(N − j)!
The first two equalities are just definitions and the third one comes from the bosonic
symmetry of v ⊗n
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
69
5. Fock-space localization and applications
We now turn to the first implication in the proof strategy (3.46). We shall need to
convert weak-∗ convergence of reduced density matrices into strong convergence, in order
to apply Theorem 3.5. The idea is to localize the state ΓN one starts from using either
compactly supported functions or finite rank orthogonal projectors. One may then work
in a compact setting with S1 -strong convergence, apply Theorem 3.5 and then pass to
the limit in the localization as a last step. More precisely, we will use localization in finite
dimensional spaces, in order to show that the general theorem can be deduced from the
finite-dimensional constructive proof discussed in the previous chapter.
The subtlelty here is that the appropriate localization notion for a N -body state (e.g. a
wave-function ΨN ∈ L2 (RdN )) is more complicated than that one is used to for one-body
wave-functions ψ ∈ L2 (Rd ). One in fact has to work directly on the reduced density matrices and localize them in such a way that the localized matrices correspond to a quantum
state. The localization procedure may lead to particle losses and thus the localizaed state
will in general not be a N -body state but a superposition of k-body states, 0 ≤ k ≤ N ,
i.e. a state on Fock space.
The localization procedure we shall use is described in Section 5.2. We shall first give
some heurtistic considerations in Section 5.1, in order to make precise what has been
said above, namely that the correct localization procedure in L2 (RdN ) must differ from
the usual localization in L2 (Rd ). Section 5.3 contains the proof of the weak quantum de
Finetti theorem and a useful auxiliary result which is a consequence of the proof using
localization.
5.1. Weak convergence and localization for a two-body state.
The following considerations are taken from [112]. Let us take a particularly simple
sequence of bosonic two-body states
1
Ψn := ψn ⊗s φn = √ (ψn ⊗ φn + φn ⊗ ψn ) ∈ L2s (R2d )
2
(5.1)
with ψn and φn being normalized in L2 (Rd ). This corresponds to having one patricle in
the state ψn and one particle in the state φn . We will assume
hφn , ψn iL2 (Rd ) = 0,
which ensures that kΨn k = 1. Extracting a subsequence if need be we have
Ψn ⇀ Ψ weakly in L2 (R2d )
and the convergence is strong if and only if kΨk = kΨn k = 1. In the case where some
mass is lost in the limit, i.e. kΨk < 1, the convergence is only weak.
We will always work in a locally compact setting and thus the only possible source for
the loss of mass is that it disappears at infinity [136, 137, 138, 139]. A possibility is that
both particles φn and ψn are lost at infinity
ψn ⇀ 0, φn ⇀ 0 in L2 (R2d ) in n → ∞,
in which case Ψn ⇀ 0 in L2 (R2d ). In L2 (R2d ) this is the scenario that is closest to the
usual loss of mass in L2 (Rd ), but there are other possibilities.
70
NICOLAS ROUGERIE
A typical case is that where only one of the two particles is lost at infinity, which we
can materialize by
ψn ⇀ 0 weakly in L2 (Rd ), φn → φ strongly in L2 (R2d ).
For the loss of mass of ψn one may typically think of the example
ψn = ψ (. + xn )
(5.2)
with |xn | → ∞ when n → ∞ and ψ say smooth with compact support. We have in this
case
Ψn ⇀ 0 in L2 (R2d )
but for obvious physical reasons we would prefer to have a weak convergence notion ensuring
1
(5.3)
Ψn ⇀g √ φ.
2
In particular, since only the parrticle in the state ψn is lost at infinity it is natural that the
limit state be one with only the particle described by φ left. We denote this convergence ⇀g
because this is precisely the geometric convergence discussed by Mathieu Lewin in [112].
The difficulty is of course that the two sides of (5.3) live in different spaces.
To introduce the correct convergence notion, one has to look at the density matrices
of Ψn :
(2)
γΨn = |φn ⊗s ψn i hφn ⊗s ψn | ⇀∗ 0 in S1 L2 (R2d )
(1)
(5.4)
γΨn = 12 |φn i hφn | + 12 |ψn i hψn | ⇀∗ 12 |φi hφ| in S1 L2 (Rd ) .
(2) (1)
One then sees that the pair γΨn , γΨn converges to the pair 0, 21 |φi hφ| that corresponds
√ −1
to the density matrices of the one-body state 2 φ ∈ L2 (Rd ). More precisely, the
geometric convergence notion is formulated in the Fock space (here bosonic with two
particles)
Fs≤2 (L2 (Rd )) := C ⊕ L2 (Rd ) ⊕ L2s (R2d )
(5.5)
1
≤2
and we have in the sense of geometric convergence on S Fs
0 ⊕ 0 ⊕ |Ψn i hΨn | ⇀g
1
2
⊕ 12 |φi hφ| ⊕ 0,
which means that all the reduced density matrices of the left-hand side converge
to those
of the right-hand side. We note that the limit does have trace 1 in S1 Fs≤2 , there is
thus no loss of mass in Fs≤2 . More precisely, in Fs≤N the loss of mass for a pure
N -particles state is materialized by the convergence to a mixed state with less
particles.
Just as the appropriate notion of weak convergence for N -body problems is different
from the usual weak convergence in L2 (RdN ) (a fortiori when taking the limit N →
∞), the appropriate procedure to localize a state and turn weak convergence into strong
convergence must be thought anew. Given a self-adjoint positive localization operator A,
say A = P a finite rank projector or A = χ the multiplication by a compactly supported
function χ, one usually localizes a wave-function ψ ∈ L2 (Rd ) by defining
ψA = Aψ
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
71
which amounts to associate
|ψi hψ| ↔ |Aψi hAψ| .
Emulating this procedure for the two-body state (5.1) one might imagine to consider a
localized state defined by its two-body density matrix
(2)
It is then clear that
γn,A = |A ⊗ AΨn i hA ⊗ AΨn | .
(2)
γn,A ⇀∗ 0 in S1 (H2s ),
which was to be expected, but it is more disturbing that we also have for the corresponding
one-body density matrix
(1)
γn,A ⇀∗ 0 in S1 (H)
whereas, in view of (5.4) one would rather like to have
1
(1)
γn,A → |Aφi hAφ| strongly in S1 (H).
2
The solution to this dilemma is to define a localized state by asking that its reduced
(2)
(1)
density matrices be A ⊗ AγΨn A ⊗ A and AγΨn A. The corresponding state is then uniquely
determined, and it so happens that it is a state on Fock space, as we explain in the next
section.
5.2. Fock-space localization.
After the preceding heuristic considerations, we now introduce the notion of localization
in the bosonic Fock space19
Fs (H) = C ⊕ H ⊕ . . . ⊕ Hns ⊕ . . .
Fs (L2 (Rd )) = C ⊕ L2 (Rd ) ⊕ . . . ⊕ L2s (Rdn ) ⊕ . . . .
(5.6)
In this course we will always start from N -body states, in which case it is sufficient to
work in the truncated Fock space
Fs≤N (H) = C ⊕ H ⊕ . . . ⊕ Hns ⊕ . . . ⊕ HN
s
Fs≤N (L2 (Rd )) = C ⊕ L2 (Rd ) ⊕ . . . ⊕ L2s (Rdn ) ⊕ . . . ⊕ L2s (RdN ).
(5.7)
Definition 5.1 (Bosonic states on the Fock space).
A bosonic state on the Fock space is a positive self-adjoint operator with trace 1 on Fs .
We denote S(Fs (H)) the set of bosonic states
S(Fs (H)) = Γ ∈ S1 (Fs (H)), Γ = Γ∗ , Γ ≥ 0, TrFs (H) [Γ] = 1 .
(5.8)
We say that a state is diagonal (stricto sensu, block-diagonal) if it can be written in the
form
Γ = G0 ⊕ G1 ⊕ . . . ⊕ Gn ⊕ . . .
(5.9)
≤N
1
n
with Gn ∈ S (Hs ). A state ΓN on the truncated Fock space Fs (H), respectively a
diagonal state on the truncated Fock space, are defined in the same manner. For a diagonal
state on Fs≤N (H), of the form
Γ = G0,N ⊕ G1,N ⊕ . . . ⊕ GN,N ,
19The procedure is the same for fermionic particles.
72
NICOLAS ROUGERIE
(n)
its n-th reduced density matrix ΓN is the operator on Hns defined by
−1 X
N k
N
(n)
ΓN =
Trn+1→k GN,k .
n
n
(5.10)
k=n
The acquainted reader will notice two things:
• We introduce only those concepts that will be crucial to the sequel of the course.
One may of course define the density matrices of general states, but we will not
need this hereafter. For a diagonal state, the reduced density matrices (5.10)
characterize the state completely. For a non-diagonal state, one must also specify
its “off-diagonal” density matrices Γ(p,q) : Hps 7→ Hqs for p 6= q.
• The normalization we chose in (5.10) is not standard. It is chosen such that, in
the spirit of the rest of the course, the n-th reduced matrix of a N - particles state
(i.e. one with G0,N = . . . = GN −1,N = 0 in (5.10)) be of trace 1. The standard
convention would rather be to fix the trace at Nn , which is less convenient to
apply Theorem 3.5. In other words the normalization takes into account the fact
that throughout the course we work we a prefered particle number N .
We may now introduce the concept of localization of a state. We shall limit ourselves
to N -body states and self-adjoint localization operators, which is sufficient for our needs
in the sequel. The following lemma/definition is taken from [112]. Other versions may be
found e.g. in [3, 55, 89].
Lemma 5.2 (Localization of a N -body state).
Let ΓN ∈ S(HN
s ) be a bosonic N -body state and A a self-adjoint operator on H with
≤N
0 ≤ A2 ≤ 1. There exists a unique diagonal state ΓA
N ∈ S(Fs (H)) such that
(n)
(n)
ΓA
(5.11)
= A⊗n ΓN A⊗n
N
for all 0 ≤ n ≤ N . Moreover, writing ΓA
N in the form
A
A
A
ΓA
N = G0,N ⊕ G1,N ⊕ . . . ⊕ GN,N ,
we have the fundamental relation
i
h √
1−A2
.
G
TrHns GA
N,n = TrHN−n
N,N
−n
s
(5.12)
Remark 5.3 (Fock-space localization).
(1) The uniqueness part of the lemma shows that one has to work on Fock space. The
localized state is in fact unique in S(Fs (H)), but to see this we would need slightly
more general definitions, cf [112].
(2) The relation (5.12) is one of the cornerstones of the method. Loosely speaking it
expresses the fact that, in the state ΓN , the probability of having n particles
√
A-localized is equal to the probability of having N − n particles 1 − A2
localized. Think of the case of a very simple localization operator, A = 1B(0,R) ,
the (multiplication by the) indicative function of the ball of radius R. We are then
simply saying that the probability of having exactly n particles in the ball equals
the probability of having exactly N − n particles outside of the ball. Indeed, in
probabilistic terms, these two possibilities correspond to the same event.
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
73
Proof of Lemma 5.2. Uniqueness, at least amongst diagonal states on the truncated Fock
space, is a simple consequence of the fact that the reduced density matrices uniquely
characterize the state. Details can be found in [112].
For the existence, one can use the usual identification (in the sense of unitary equivalence)
p
p
Fs (AH ⊕ 1 − A2 H) ≃ Fs (AH) ⊗ Fs ( 1 − A2 H)
and define the localized state by taking a partial trace with respect to the second Hilbert
space in the tensor product of the right-hand side. We shall follow a more explicit but
equivalent route. To simplify √
notation we shall restrict to the case where A = P is an
orthogonal projector and thus 1 − A2 = P⊥ .
First note that for any n-body observable On
i
i
h
h
(n)
(n)
Tr On P ⊗n ΓN P ⊗n = Tr P ⊗n On P ⊗n ΓN
= Tr (P ⊗n On P ⊗n ) ⊗ 1⊗N −n ΓN
= Tr On ⊗ 1⊗N −n (P ⊗n ⊗ 1⊗N −n ΓN P ⊗n ⊗ 1⊗N −n )
so that
(n)
P ⊗n ΓN P ⊗n = Trn+1→N P ⊗n ⊗ 1⊗N −n ΓN P ⊗n ⊗ 1⊗N −n .
Thus, by cyclicity of the trace, we obtain
i
h
(n)
P ⊗n ΓN P ⊗n = Trn+1→N (P ⊗n ⊗ 1⊗(N −n) )P ⊗n ⊗ 1⊗(N −n) ΓN P ⊗n ⊗ 1⊗(N −n)
N
−n i
h
X
N −n
⊗(N −n−k)
=
ΓN P ⊗n ⊗ 1⊗(N −n)
Trn+1→N P ⊗n+k ⊗ P⊥
k
k=0
N
−n i
h
X
N −n
⊗(N −n−k)
⊗(N −n−k)
ΓN P ⊗n+k ⊗ P⊥
Trn+1→N P ⊗n+k ⊗ P⊥
=
k
k=0
N i
h
X
N −n
⊗(N −k)
⊗(N −k)
.
ΓN P ⊗k ⊗ P⊥
Trn+1→N P ⊗k ⊗ P⊥
=
k−n
k=n
Here we inserted 1 = P +P⊥ in the first occurence of 1⊗N −n in the first line and expanded.
Since the state P ⊗n ⊗ 1⊗(N −n) ΓN P ⊗n ⊗ 1⊗(N −n) we act on is symmetric under exchange
of the last N − n variables, we can reorganize the terms in the expansion to obtain the
sum of the second line, involving binomial coefficients. It then suffices to note that
−1 N −n
N
N
k
=
k−n
k
n
n
to deduce
(n)
P ⊗n ΓN P ⊗n
=
N −1 X
N
k
k=n
n
n
(n)
Trn+1→N GPN,k = GPN
with (cf Definition (5.10))
h
i
N
⊗(N −k)
⊗(N −k)
P
GN,k =
Trk+1→N P ⊗k ⊗ P⊥
ΓN P ⊗k ⊗ P⊥
k
(5.13)
74
NICOLAS ROUGERIE
and
GPN = GPN,0 ⊕ . . . ⊕ GPN,N .
This is indeed an operator on Fs≤N (P H): in (5.13), one can interchange the first k particles
and they live on P H, whereas the last N − k particles are traced out.
There remains to show that GPN is indeed a state, i.e. that its trace is 1. To see this,
we write
i
h
1 = TrHN [ΓN ] = TrHN (P + P⊥ )⊗N ΓN (P + P⊥ )⊗N
N h
i
X
N
⊗(N −k)
⊗(N −k)
ΓN P ⊗k ⊗ P⊥
TrHN P ⊗k ⊗ P⊥
=
k
=
k=0
N
X
k=0
TrHk GPN,k = TrF (H) [GPN ].
The relation (5.12) is an immediate consequence of (5.13) and the symmetry of ΓN .
5.3. Proof of the weak quantum de Finetti theorem and corollaries.
We are now going to use the localization procedure just described to prove the first
implication of the strategy (3.46). The idea is to use a finite-rank projector P , and then
combine (5.10) with (5.11) to write (with n ∈ N fixed)
N −1 X
N
k
⊗n (n) ⊗n
=
P γN P
Trn+1→k GPN,k
n
n
k=n
N X
k n
≈
(5.14)
Trn+1→k GPN,k .
N
k=n
Here we inserted the simple estimate (see the computation in [113], Equation (2.13))
−1 n
N
k
k
+ O(N −1 ).
(5.15)
=
N
n
n
We then argue as follows: The terms
n where k is small contribute very little to the
sum (5.14) because of the factor Nk . For the terms where k is large, we note that,
up to normalization, GPN,k is a k-particles bosonic state over P H. One may thus apply
the de Finetti theorem discussed in Chapter 4 to it, without worrying about compactness
issues since P H has finite dimension. Since k is large in these terms, and n is fixed, we
obtain (formally)
Z
dνk (u)|u⊗n ihu⊗n |
Trn+1→k GPN,k ≈ TrHk [GPN,k ]
u∈SP H
for a certain measure νk , and thus
(n)
P ⊗n γN P ⊗n
≈
N
X
k≃N
TrHk [GPN,k ]
k
N
n Z
u∈SP H
dνk (u)|u⊗n ihu⊗n |.
In the limit N → ∞, the discrete sum should become an integral in λ = k/N . Using the
P
P
fact that GPN is a state to deal with normalization (recall that N
k=1 Tr[GN,k ] = 1), it is
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
natural to hope that we can obtain
Z
⊗n (n) ⊗n
≈
P γN P
1
λ
0
n
Z
u∈SP H
75
dν(λ, u)(u)|u⊗n ihu⊗n |,
which may be rewritten in the form (3.24) by defining (in “spherical” coordinates on the
unit ball BP H)
u
u
2
:= d kuk × dνkuk2
.
dµ(u) = dµ kuk ,
kuk
kuk
There remains, as a last step, to apply this procedure for a sequence of projectors Pℓ → 1
and to check some compatibility relations to conclude. The final measure might not be a
probability measure, which one can compensate by adding a delta function at the origin
without changing any of the previous formulae.
Note that this proof, which combines the methods of Chapter 4 and Section 5.2, gives
a recipe to construct the de Finetti measure “by hand” (up to the fact that we have to
pass to the limit at some point). This is very useful in practice, see Chapters 6 and 7.
The spirit of the proof using localization is reminiscent of some aspects of the method of
Ammari and Nier [4, 5] who define cylindrical projections of the measure first.
Let us now present the details of the proof, following [113, Section 2].
Proof of Theorem 3.7. Existence. We carry on with the above notation. The following
(n)
formula defines MP,N as a measure on [0, 1], with values in the positive hermitian matrices
of size dim (⊗ns (P H)):
N
X
(n)
dMP,N (λ) :=
δk/N (λ) Trn+1→k GPN,k .
k=n
We then have, using (5.15)
Z 1
N
⊗n (n) ⊗n
C X
(n)
n
Tr GPN,k → 0 when N → ∞.
Tr P γΓN P −
λ dMP,N (λ) ≤
N
0
(5.16)
k=n
Since P is a finite rank projector and
(n)
γN
converges weakly-∗ by assumption,
(n)
P ⊗n γN P ⊗n → P ⊗n γ (n) P ⊗n
(5.17)
strongly in trace-class norm. On the other hand
Z 1
N
N
X
X
(n)
dMP,N (λ) =
TrHk GPN,k ≤
TrHn
TrHk GPN,k = 1,
0
k=n+1
k=0
(n)
MP,N
is a sequence of measures with bounded total variation over a compact finite
so
dimensional space (positive hermitian matrices of size dim P H having a trace less than
(n)
1). One may thus extract from it a subsequence converging weakly as measures to MP .
Combining with (5.16) and (5.17) we have
Z 1
(n)
⊗n (n) ⊗n
λn dMP (λ).
(5.18)
P γ P
=
0
76
NICOLAS ROUGERIE
(n)
We now have to show that the sequence of measures MP
n∈N
that we just obtained
is consistent in the sense that, for all n ≥ 0 and for all continuous functions f over [0, 1]
vanishing at 0,
Z 1
Z 1
(n)
(n+1)
(5.19)
f (λ)dMP (λ).
(λ) =
f (λ) Trn+1 dMP
0
0
Here Trn+1 denotes partial trace with respect to the last variable. We have
N
X
(n+1)
Trn+1 dMP,N (λ) =
δk/N (λ) Trn→k GN,k
k=n+1
(n)
= dMP,N (λ) − δn/N (λ)GPN,n
and thus
Z 1
0
Z
(n)
(n+1)
f (λ) TrHn Trn+1 dMP,N (λ) − dMP,N (λ) ≤
≤f
1
f (λ)δn/N (λ) TrHn GPN,n
n
0
N
since TrHn GPN,n ≤ 1. Passing to the limit we obtain (5.19) for all function f such that
f (0) = 0.
We now apply Theorem 3.5 in finite dimension. Stricto sensu, this result applies only
(n)
at fixed λ, but approaching dMP by step functions and then passing to the limit, we
obtain a measure νP on [0, 1] × SH ∩ (P H) such that
Z Z 1
Z 1
(n)
f (λ)dνP (λ, u)|u⊗n ihu⊗n |
f (λ)dMP (λ) =
SH
0
0
for all continuous functions f vanishing at 0. We thus obtain
Z 1Z
⊗n (n) ⊗n
dνP (λ, u) λn |u⊗n ihu⊗n |
P γ P
=
0
=
=
Z
Z0
1Z
BH
SH
SH
√
√
dνP (λ, u) |( λu)⊗n ih( λu)⊗n |
dµP (u)|u⊗n ihu⊗n |,
defining the measure µP in spherical coordinates. We are free to add a Dirac mass at the
origin to turn µP into a probability measure.
The argument can be applied to a sequence of finite rank projectors converging to the
identity. We then have a sequence of probability measures µk on BH such that
Z
dµk (u) |u⊗n ihu⊗n |.
Pk⊗n γ (n) Pk⊗n =
BH
Taking an increasing sequence of projectors (i.e. Pk H ⊂ Pk+1 H) it is clear that µk coı̈ncides
with µℓ on Pℓ H for ℓ ≤ k (cf the argument for measure uniqueness below). Since all these
measures have their supports in a bounded set, there exists (see for example [186, Lemma
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
77
1]) a unique probability measure20 µ on BH which coı̈ncides with µk on Pk H in the sense
that:
Z
Z
⊗n
⊗n
dµ(u) |(Pk u)⊗n ih(Pk u)⊗n |.
dµk (u) |u ihu | =
BH
BH
We thus conclude
Pk⊗n γ (n) Pk⊗n = Pk⊗n
Z
BH
dµ(u) |u⊗n ihu⊗n | Pk⊗n
and there only remains to take the limit k → ∞ to deduce the existence of a measure
satisfying (3.24).
Uniqueness. Let us now prove that the measure just constructed must be unique. Let
µ and µ′ satisfy
Z
Z
⊗k
⊗k
|u⊗k ihu⊗k |dµ′ (u) = 0
(5.20)
|u ihu |dµ(u) −
BH
BH
for all k ≥ 1. Let V = vect(e1 , . . . , ed ) be a finite-dimensional subspace of H, Pi =
|ei ihei | the associated projectors and µV , µ′V the cylindrical projections of µ and µ′ over V .
Applying Pi1 ⊗ · · · ⊗ Pik on the left and Pj1 ⊗ · · · ⊗ Pjk on the right to (5.20), we obtain
Z
ui1 · · · uik uj1 · · · ujk d(µV − µ′V )(u) = 0
BV
P
for all mutli-indices i1 , ..., ik and j1 , ..., jk (we denote u = dj=1 uj ej ). On the other hand,
by S 1 invariance of both measures it is clear that if k 6= ℓ,
Z
ui1 · · · uik uj1 · · · ujℓ d(µV − µ′V )(u) = 0.
BV
Since polynomials in ui and uj are dense in C 0 (BV, C) (continuous functions on the unit
ball of V ), we deduce that the cylindrical projections of µ and µ′ over V coı̈ncide. That
being true for all finite dimensional subspace V , we conclude that the two measures must
coı̈ncide everywhere.
We now give a particular case, adapted to our needs in the next chapter, of a very useful
corollary of the above method of proof (see [113, Theorem 2.6] for the general statement):
Corollary 5.4 (Localization and de Finetti measure).
Let (ΓN )N ∈N be a sequence of N -body states over H = L2 (Rd ) satisfying the assumptions
of Theorem 3.7 and µ be the associated de Finetti measure. Assume that
i
i
h
h
(1)
(1)
(5.21)
Tr −∆γN = Tr |∇|γN |∇| ≤ C
for some constant independent of N . Let χ be a localization function with compact support
with 0 ≤ χ ≤ 1 and GχN be the localized state defined in Lemma 5.2. Then
Z
N
X
k
χ
dµ(u) f (kχuk2 )
(5.22)
TrHk GN,k =
f
lim
N →∞
N
BH
k=0
for all continuous functions f on [0, 1].
20To construct it, note that the σ-closure of the union for k ≥ 0 of the borelians of P H coı̈ncides with
k
the borelians of H.
78
NICOLAS ROUGERIE
Proof. Since polynomials are dense in the continuous functions on [0, 1] it is sufficient to
consider the case f (λ) = λn with n = 0, 1, .... We then use (5.10), (5.11) and (5.15) again
to write
N X k n
i
h
(n)
χ
TrHk GN,k − TrHn χ⊗n γN χ⊗n → 0 when N → ∞.
N
k=0
Assumption (5.21) ensures

Tr 
n
X
j=1


(n)
−∆j  γN  ≤ Cn
and we may thus assume that








n
n
n
n
X
X
X
X
(n)

|∇|j  γN 
|∇|j  ⇀∗ 
|∇|j  γ (n) 
|∇|j  when N → ∞.
j=1
j=1
j=1
j=1
The multiplication operator by χ is relatively compact with respect to the Laplacian, thus

−1
n
X
Dnχ := χ⊗n 
|∇|j 
j=1
is a compact operator. We then have
h
(n)
i




TrHn χ⊗n γN χ⊗n = TrHn Dnχ 
n
X
j=1
→ TrHn Dnχ 
h




n
X
(n)
|∇|j  γN 
|∇|j  Dnχ 
n
X
j=1

j=1

|∇|j  γ (n) 
i
= TrHn χ⊗n γ (n) χ⊗n .
n
X
j=1


|∇|j  Dnχ 
We conclude by using (3.24) that
Z
N X
k n
dµ(u) TrHn |(χu)⊗n ih(χu)⊗n |
TrHk GχN,k →
N
BH
k=0
Z
Z
2n
dµ(u)f (kχuk2 ).
dµ(u) kχuk =
=
BH
BH
Remark 5.5 (Weak de Finetti measure and loss of mass.).
(1) Assumption (5.21) is used to ensure strong compactness of density matrices, see the
proof. It is of course very natural for the applications we have in mind (uniformly
bounded kinetic energy). The convergence in (5.22) means that the mass of the de
Finetti measure µ on the sphere {kuk2 = λ} corresponds to the probability that a
fraction λ of the particles does not escape to infinity.
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
79
(2) We will use this corollary to obtain information on particles escaping to infinity in
the following manner. We define the function
p
η = 1 − χ2
which localizes “close to infinity”. One of course cannot apply the result directly
to the localized state GηN . Instead one may use Relation (5.12) to obtain
N
N
X
X
k
k
η
f 1−
TrHk GN,k = lim
TrHk GχN,k
f
lim
N →∞
N →∞
N
N
k=0
k=0
Z
dµ(u) f 1 − kχuk2 ,
(5.23)
=
BH
which gives some control on the loss of mass at infinity, encoded in the de Finetti
measure. This is a bit surprising since the latter by definition describes particles
which stay trapped. Typically we will use (5.23) with f (λ) ≃ eH (λ), the Hartree
energy at mass λ, see the next chapter.
80
NICOLAS ROUGERIE
6. Derivation of Hartree’s theory: general case
We now turn to the general case of the derivation of Hartree’s functional as a limit of the
N -body problem in the mean-field regime. In Chapter 3 we saw that, under simplifying
physical assumptions, the result is a rather direct consequence of the weak and strong de
Finetti theorems. The general case requires a more thorough analysis and we shall use
fully the localization tools introduced in Chapter 5.
The setting is now that of particles in a non-trapping external potential V , interacting
via a potential that may have bound states. Let us recap the notation: The Hamiltonian
of the full system is
V
HN
=
N
X
Tj +
j=1
acting on the Hilbert space HN
s =
NN
s
1
N −1
X
1≤i<j≤N
w(xi − xj ),
(6.1)
H, with H = L2 (Rd ). The one-body Hamiltonian is
T = −∆ + V,
(6.2)
that we assume to be self-adjoint and bounded below. We have emphasized the dependence
on the potential V in the notation (6.1) because we will be lead to consider the system
0 where one sets V ≡ 0.
where particles are lost at infinity described by the Hamiltonian HN
One can generalize in the same directions as mentioned in Remark 3.2, but we shall for
simplicity stick to the above model case.
The interaction potential w : R 7→ R will be asumed bounded relatively to T : for some
0 ≤ β− , β+ ≤ 1 and C > 0
− β− (T1 + T2 ) − C ≤ w(x1 − x2 ) ≤ β+ (T1 + T2 ) + C.
(6.3)
We also assume symmetry
w(−x) = w(x),
and some decay at infinity
w ∈ Lp (Rd ) + L∞ (Rd ), max(1, d/2) < p < ∞ → 0, w(x) → 0 when |x| → ∞.
(6.4)
Again, it is rather vain to consider partially trapping one-body potentials, and we thus
assume that V is non-trapping in all directions:
V ∈ Lp (Rd ) + L∞ (Rd ), max 1, d/2 ≤ p < ∞, V (x) → 0 when |x| → ∞.
(6.5)
This ensures [160] that HN is self-adjoint and bounded below. The ground state energy
of (3.1) is always given by
V
V
=
inf
E V (N ) = inf σHN HN
Ψ, HN
Ψ HN .
(6.6)
Ψ∈HN ,kΨk=1
Finally let us recall what the limit objects are. The Hartree functional with potential V
is given by
ZZ
Z
1
2
2
V
|u(x)|2 w(x − y)|u(y)|2 dxdy
(6.7)
|∇u| + V |u| +
EH [u] :=
2
Rd ×Rd
Rd
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
81
and we shall use the notation EH0 for the translation-invariant functional where V ≡ 0.
The Hartree energy at mass λ is given by
eVH (λ) :=
inf EHV [u],
kuk2 =λ
0 ≤ λ ≤ 1.
(6.8)
Under the previous assumptions we will always have the binding inequality
eVH (1) ≤ eVH (λ) + e0H (1 − λ)
(6.9)
which is easily proved by evaluating the energy of a sequence of functions with a mass λ
in the well of the potential V and a mass 1 − λ escaping to infinity. We will prove the
following theorem, extracted from [113] (particular case of Theorem 1.1 therein).
Theorem 6.1 (Derivation of Hartree’s theory, general case).
Under the preceding assumptions, we have:
(i) Convergence of energy.
E V (N )
= eVH (1).
N →∞
N
lim
(6.10)
(ii) Convergence of states. Let ΨN be a sequence of L2 (RdN )-normalized quasi-minimizers
V:
for HN
V
ΨN , HN
ΨN = E V (N ) + o(N ) when N → ∞,
(6.11)
(k)
and γN be the corresponding reduced density matrices. There exists µ ∈ P(BH) a probability measure on the unit ball of H with µ(MV ) = 1, where
n
o
MV = u ∈ BH : EHV [u] = eVH (kuk2 ) = eVH (1) − e0H (1 − kuk2 ) ,
(6.12)
such that, along a subsequence,
(k)
γNj
⇀∗
Z
MV
|u⊗k ihu⊗k | dµ(u)
(6.13)
weakly-∗ in S1 (Hk ), for all k ≥ 1.
(iii) If in addition the strict binding inequality
eVH (1) < eVH (λ) + e0H (1 − λ)
(6.14)
⊗k
1
k
γN → |u⊗k
H ihuH | strongly in S (H )
(6.15)
holds for all 0 ≤ λ < 1, the measure µ has its support in the sphere SH and the limit (6.13)
holds in trace-class norm. In particular, if eVH (1) has a unique (modulo a constant phase)
minimizer uH , then for the whole sequence
(k)
for all fixed k ≥ 1.
Remark 6.2 (Characterization of the limit set).
It is a classical fact that
n
o
MV = u ∈ BH | ∃(un ) minimizing sequence for eVH (1) such that un ⇀ u weakly in L2 (Rd ) .
This follows from the usual concentration compactness method for non-linear one-body
variational problems. One can thus interpret (6.13) as saying that the weak limits for
82
NICOLAS ROUGERIE
the N -body problem are fully characterized in terms of the weak limits for the one-body
problem.
The proof proceeds in two steps. In Section 6.1 we first consider the completely
translation-invariant case where V ≡ 0, which will describe particles escaping from the potential well V in the general case. We show that the energy e0H (1) is the limit of N −1 E 0 (N ).
In this case one cannot hope for much more since there always exist minimizing sequences
whose density matrices converge to 0 in trace-class norm. In addition to the tools already
introduced we shall rely on ideas of Lévy-Leblond, Lieb, Thirring and Yau [111, 132, 133]
to recover a bit of compactness.
We then use fully the localization methods of Chapter 5 to treat the general case
in Section 6.2. In the spirit of the concentration-compactness principle we will localize
minimizing sequences inside and outside of a ball. The inside-localized part is described
by the weak-∗ limit of reduced density matrices and we can thus use the weak de Finetti
theorem. The outside-localized part no longer sees the potential V and we may thus apply
to it the results of Section 6.1.
6.1. The translation-invariant problem.
Here we deal with the case where V ≡ 0. It is then possible to construct quasi(1)
minimizing sequences for E 0 (N ) with γN (. − yN ) ⇀∗ 0 for any sequence of translations
xN . One may thus have vanishing in Lions’ terminology [136, 137] and without any specific
trick one cannot hope for more than the convergence of the energy. Indeed, it is possible
to construct a state where the relative motion of the particles is bound by the interaction
potential w but where the center of mass vanishes. This implies vanishing for the whole
sequence. We shall thus be content with proving the convergence of the energy:
Theorem 6.3 (Energy of translation-invariant systems).
Under the preceding assumptions, we have
E 0 (N )
= e0H (1).
N →∞
N
lim
(6.16)
The first step is to use part of the interaction potential to create an attractive one-body
potential (this roughly amounts to taking out the center of mass degree of freedom). This
way we will define an auxiliary problem whose energy is close to the original one, but
for which particles always stay trapped. This is done in the following lemma, inspired
from [132, 133]:
Lemma 6.4 (Auxiliary problem with binding).
We split w = w+ − w− into positive and negative parts and define, for some ε > 0
wε (x) = w(x) + εw− (x).
Consider the auxiliary Hamiltonian
N X
ε
Ki − εw− (xi ) +
HN =
i=1
1
N −1
and E ε (N ) the associated ground state energy. Then
X
1≤i<j≤N
wε (xi − xj )
E 0 (N )
E ε (N )
≤ lim
=: a.
N →∞
N →∞
N
N
aε := lim
(6.17)
(6.18)
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
83
Proof. Using symmetry, we can for all Ψ ∈ L2s (RdN ) write
! +
! +
*
*
N
N
−1
X
X
N
−∆i Ψ =
−∆i Ψ .
Ψ,
Ψ,
N −1
i=1
Similarly
2
N (N − 1)
*
Ψ,
X
1≤i<j≤N
w(xi − xj )Ψ
with γΨ = |ΨihΨ|. Then
h
(2)
TrH2 wε γΨ
and
ε TrH2
All this implies
i=1
i
h
+
i
i
i
h
h
h
(2)
(2)
(2)
= TrH2 wγΨ = TrH2 wε γΨ − ε TrH2 w− γΨ

2
=
Tr 
(N − 1)(N − 2)
1≤i<j≤N −1
(2)
w− γΨ
i

wε (xi − xj )γΨ 
#
"N −1
X
ε
−
Tr
w (xi − xN )γΨ .
=
N −1
N −1
0
hΨ, HN
Ψi
N
* N −1
X
− ∆i − εw− (xi − xN ) +
= Ψ, 
i=1
X
i=1
1
N −2
X
1≤i<j≤N −1
 +
wε (xi − xj ) Ψ . (6.19)
In the preceding equation the Hamiltonian between parenthesis depends on xN via the
one-body potential εw− (xi − xN ) but since the other terms are translation-invariant the
bottom of the spectrum is in fact independent of xN . We thus have
N −1
0
hΨ, HN
Ψi ≥ E ε (N − 1)hΨ, Ψi
(6.20)
N
for all Ψ ∈ L2s (RdN ), which implies
E ε (N − 1)
E 0 (N )
≥
.
N
N −1
The sequences E 0 (N )/N and E ε (N )/N are increasing since they are the infimum of variational problems set on decreasing sets (cf (1.55)). Using trial states spread over larger
and larger domains method, one may on the other hand show that
E 0 (N )
E ε (N )
≤ 0,
≤0
N
N
and thus that the limits a and aε exist. Then (6.20) clearly implies (6.18).
We next derive a lower bound to E ε (N ) for ε small enough. It is much easier to work on
this energy because the corresponding sequences of reduced density matrices are strongly
compact in S1 . Indeed, compactness or its absence (physically, binding or its absence)
resuts from a comparison between the attractive and repulsive parts of the one- and twobody potentials. Here the one-body potential εw− and the two-body potential wε are
84
NICOLAS ROUGERIE
well equilibrated because they have been built precisely for this purpose, starting from the
original two-body potential w. This trick will allow us to conclude the
Proof of Theorem 6.3. As usual, only the lower bound is non-trivial. Since e0H (1) ≤ 0 one
may assume that a < 0 since otherwise a = e0H (1) = 0 and there is nothing to prove. We
are going to prove the lower bound
ZZ
1
ε
2
2
aε ≥ eH (1) := inf
hu, (−∆ − εw− )ui +
|u(x)| wε (x − y)|u(y)| dxdy , (6.21)
2
kuk2 =1
and it is then easy to show that eεH (1) → e0H (1) when ε → to obtain (6.16) by combining
with (6.18).
Let ΨN be a sequence of wave-functions such that
ε
hΨN , HN
ΨN i = E ε (N ) + o(N ).
(k)
and γN the corresponding reduced density matrices. Then
i 1
i
h
h
ε Ψ i
hΨN , HN
(1)
(2)
N
= lim TrH (−∆ − εw− )γN + TrH2 wε γN
.
aε = lim
N →∞
N →∞
N
2
After the usual diagonal extraction we can assume that
(k)
γN ⇀∗ γ (k)
and we are going to show that the convergence is actually strong. We shall afterwards use
the strong de Finetti theorem to obtain Hartree’s energy as a lower bound.
2 = 1 and use Lemma 3.12 to get
We pick a smooth partition of unity χ2R + ηR
i
h
i 1
h
(2) ⊗2
(1)
χ
γ
aε ≥ lim inf lim inf TrH (−∆ − εw− )χR γN χR + TrH2 wε χ⊗2
R
R N
R→∞ N →∞
2
i
h
i 1
h
(1)
⊗2 (2) ⊗2
γN ηR
. (6.22)
+ TrH −∆ηR γN ηR + TrH2 wε ηR
2
We define the χR – and ηR –localized states GχN and GηN by applying Lemma 5.2 to ΨN .
We will use those to estimate separately the two terms in the right side of (6.22).
The χR -localized term. Using (5.10) and (5.11) we have
i
i 1
h
h
(2) ⊗2
(1)
χ
γ
TrH (−∆ − εw− )χR γN χR + TrH2 wε χ⊗2
R
R N

 2

N
k
k
X
X
X
1
1
TrHk  (−∆ − εw− )i +
wε (xi − xj ) GχN,k  (6.23)
=
N
N −1
k=1
i<j
i=1
We apply the inequality
A + tB = (1 − t)A + t(A + B) ≥ (1 − t) inf σ(A) + t inf σ(A + B)
with
k
X
A=
(−∆ − εw− )ℓ ,
ℓ=1
A + B = Hε,k ,
t = (k − 1)/(N − 1).
(6.24)
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
85
We have
lim inf σ(−∆ − εw− ) = inf σ(−∆) = 0
ε→0
and since we assume that a < 0, for ε small enough
inf σ(−∆ − εw− ) > a ≥ aε ≥ k−1 inf σ(Hε,k ).
Thus
inf σ(A) ≥ inf σ(A + B)
and we may then write
N
i X
h
i 1
h
k Tr GχN,k E ε (k)
(2) ⊗2
(1)
≥
χ
.
γ
TrH (−∆ − εw− )χR γN χR + TrH2 wε χ⊗2
R
R N
2
N
k
k=1
But since
N
X
k Tr GχN,k
k=0
and k 7→
E ε (k)
k
N
i
i
h
h
(1)
−→ Tr χ2R γ (1)
= Tr χ2R γN
N →∞
and
E ε (k)
= aε ,
k→∞
k
lim
is increasing we conclude
i
h
i
h
i 1
h
(1)
⊗2 (2) ⊗2
≥ aε Tr χ2R γ (1)
lim inf TrH (−∆ − εw− )χR γN χR + TrH2 wε χR γN χR
N →∞
2
(6.25)
by monotone convergence.
The ηR -localized term. Using the results of Section 5.2 as above we have
i 1
h
i
h
(1)
⊗2 (2) ⊗2
TrH T ηR γN ηR + TrH2 wε ηR
γN ηR
2



N
k
k
X
X
1 X
1
wε (xi − xj ) GηN,k  . (6.26)
=
TrHk 
−∆i +
N
N −1
k=1
i<j
i=1
Here we use
to obtain
k
X
i=1
−∆ ≥ 0,
wε = w + 2εw− ≥ (1 − 2ε)w and E 0 (k) ≤ ak < 0
k
1 X
−∆i +
wε (xi − xj ) ≥
N −1
i<j
(1 − 2ε)(k − 1) 0
Hk
N −1
(1 − 2ε)(k − 1) 0
E (k) ≥ E 0 (k) − 2εak
N −1
for all k ≥ 1. Combining with (6.26) we obtain
0
N
i 1
h
i X
h
k Tr GηN,k
E (k)
(1)
⊗2 (2) ⊗2
TrH −∆ηR γN ηR + TrH2 wε ηR γN ηR ≥
·
− 2εa .
2
N
k
≥
k=1
We finally deduce that
i
h
i 1
h
(1)
⊗2 (2) ⊗2
≥ (1−2ε)a 1−Tr[χ2R γ (1) ] (6.27)
lim inf TrH T ηR γN ηR + TrH2 wε ηR γN ηR
N →∞
2
86
NICOLAS ROUGERIE
upon using
N
i
i
h
h
X
k
2 (1)
−→ 1 − Tr χ2R γ (1)
Tr GηN,k = Tr ηR
γN
N →∞
N
k=0
as well as the fact that k 7→
E 0 (k)
k
and
E 0 (k)
=a
k→∞
k
lim
is increasing.
Conclusion. Inserting (6.25) and (6.27) in (6.22) we find
i i
h
h
aε ≥ lim inf aε Tr χ2R γ (1) + (1 − 2ε)a 1 − Tr χ2R γ (1)
R→∞
= aε Tr[γ (1) ] + (1 − 2ε)a 1 − Tr[γ (1) ]
Since we assumed aε ≤ a < 0, we obtain
Tr[γ (1) ] = 1.
(6.28)
There is thus no loss of mass for the auxiliary problem defined in Lemma 6.4. This suffices
(k)
to conclude that the reduced density matrices γN converge strongly. Indeed, one may
apply the weak de Finetti theorem to the sequence of weak limits γ (k) to obtain a measure
µ living on the unit ball of H. But (6.28) combined with (3.24) implies that the measure
must in fact live on the unit sphere. We thus have, for all k ≥ 0,
Tr[γ (k) ] = 1,
(k)
which implies that the convergence of γN to γ (k) is actually strong in trace-class norm.
We can then go back to (6.22) to obtain
i
i 1
h
h
(1)
(2)
lim inf TrH (−∆ − εw− )γN + TrH2 wε γN
N →∞
2
h
i 1
h
i
≥ TrH (−∆ − εw− )γ (1) + TrH2 wε γ (2) .
2
We then apply the strong de Finetti theorem to the limits of the reduced density matrices
to conclude that the right side is necessarily larger than eεH (1). This gives (6.21) and
concludes the proof.
6.2. Concluding the proof in the general case.
We have almost all the ingredients of the proof of Theorem 6.1 at our disposal. We
only need a little bit more information on the translation-invariant problem, as we now
explain.
For k ≥ 2 consider the energy


k
k
X
X
λ
1
(6.29)
w(xi − xj ) ,
−∆i +
bk (λ) := inf σHk 
k
k−1
i=1
i<j
λ
i.e. an energy for k particles with an interaction strength proportional to k−1
. In the last
section we have already shown that the limit k → ∞ of such an energy is given by λe0H (λ)
when λ is a fixed parameter. Using Fock-space localization methods, the energy of particles
lost at infinity in the minimization of the energy for N ≥ k particles will be naturally
described as a superposition of energies for k-particles systems with an interaction of
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
87
strength 1/(N − 1) inherited from the original problem. In other words, we will have to
evaluate a superposition of the energies bk (λ) with
k−1
,
λ=
N −1
a bit as in (6.23) and (6.26). Since this λ depends on k it will be useful to know that bk (λ)
is equicontinuous as a function of λ:
Lemma 6.5 (Equi-continuity of the energy as a function of the interaction).
We set the convention
b0 (λ) = b1 (λ) = 0.
Then, for all λ ∈ [0, 1]
lim λ bk (λ) = e0H (λ).
(6.30)
k→∞
Moreover, for all 0 ≤ λ ≤ λ′ ≤ 1
0 ≤ bk (λ) − bk (λ′ ) ≤ C|λ − λ′ |
(6.31)
where C does not depend on k.
Proof. We start by vindicating our claim that (6.30) is a direct consequence of the analysis
of the previous section. For λ = 0 there is nothing to prove. For λ > 0 we use Theorem 6.3
to obtain
ZZ
λ
2
2
lim λ bk (λ) = λ inf
hu, Kui +
w(x − y)|u(x)| |u(y)|
k→∞
2
kuk2 =1
ZZ
1
w(x − y)|u(x)|2 |u(y)|2 = e0H (λ).
= inf
hu, Kui +
2
kuk2 =λ
The fact that
bk (λ) ≥ bk (λ′ ) for all 0 ≤ λ < λ′ ≤ 1
is a consequence of (6.24). Indeed, one can see that either bk (λ) = 0 for all λ or the same
kind of argument as in (6.24) applies.
For the equicontinuity (6.31) we fix some 0 < α ≤ 1 and remark that, with
we have
δ := (λ′ − λ)(α−1 − λ)−1 ,




k
k
k
k
X
X
1 X
λ′ X
λ
1
−
δ

Ki +
wij  =
Ki +
wij 
k
k−1
k
k−1
i=1
i<j
i=1
i<j


k
k
1 X
δ  X
Ki +
wij 
α
+
kα
k−1
i=1
i<j
Cδ
α
using the fact the spectrum of the operator appearing in the second line is bounded below.
We deduce
0 ≤ bk (λ) − bk (λ′ ) ≤ δ(bk (λ) + Cα−1 ) ≤ C|λ′ − λ|
since bk (λ) is uniformly bounded and |δ| ≤ C|λ − λ′ |.
≥ (1 − δ)bk (λ) −
88
NICOLAS ROUGERIE
We may know conclude the
Proof of Theorem 6.1. Let ΨN be a sequence of N -body wave-functions such that
V
hΨN , HN
ΨN i = E V (N ) + o(N ),
(k)
with γN the associated reduced density matrices. After a diagonal extraction we have
(k)
γN ⇀ γ (k)
weakly-∗. Theorem 3.7 ensures the existence of a probability measure µ such that
Z
dµ(u)|u⊗k ihu⊗k |.
(6.32)
γ (k) =
u∈BH
2 = 1 as
We pick a smooth partition of unity χ2R + ηR
χ
η
state GN and GN constructed from |ΨN ihΨN |. Using
previously and define the localized
Lemma 3.12 again we obtain
i
i
h
h
1
E V (N )
(1)
(2)
= lim TrH T γN + TrH2 wγN
lim
N →∞
N →∞
N
2
i 1
h
i
h
(1)
(2) ⊗2
≥ lim inf lim inf TrH T χR γN χR + TrH2 wχ⊗2
χ
γ
R
R N
R→∞ N →∞
2
i
h
i 1
h
(1)
⊗2 (2) ⊗2
γN ηR
.
(6.33)
+ TrH −∆ηR γN ηR + TrH2 wηR
2
First we use the strong local compactness and (6.32) for the χR -localized term:
i
h
i 1
h
(1)
⊗2 (2) ⊗2
lim inf TrH T χR γN χR + TrH2 wχR γN χR
N →∞
2
i 1
h
i Z
h
⊗2 (2) ⊗2
(1)
EHV [χR u]dµ(u). (6.34)
≥ TrH T χR γ χR + TrH2 wχR γ χR =
2
BH
Our main task is to control the second term in the right side of (6.33). We claim that
h
i 1
i Z
h
(1)
⊗2 (2) ⊗2
lim inf Tr T ηR γN ηR + TrH2 wηR γN ηR
e0H (1 − kχR uk2 )dµ(u). (6.35)
≥
N →∞
2
BH
Indeed, using the ηR -localized state GηN we have
h
(1)
TrH −∆ηR γN ηR
i



N
k
k
X
X
X
1
1
1
⊗2 (2) ⊗2
wij  GηN,k 
γN ηR =
−∆i +
TrHk 
+ TrH2 wηR
2
N
N −1
i<j
i=1
k=1


N
k
k
X
X
1
1 X
≥
Tr GηN,k inf σHk 
wij 
−∆i +
N
N −1
i
h
i=1
k=1
≥
N
X
Tr GηN,k
k=0
k
bk
N
k−1
N −1
i<j
where bk is the function defined in (6.29). On the other hand
N
X
k
k−1
k
η
0
lim
Tr GN,k
bk
− eH
= 0,
N →∞
N
N −1
N
k=0
(6.36)
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
89
0
because, using the equicontinuity of {bk }∞
k=1 and the convergence limk→∞ λbk (λ) = eH (λ)
given by Lemma 6.5, we get
k
k−1
k 0
lim
sup bk
= 0.
− eH
N →∞
N
N −1
N k=1,2,...,N
We just have to combine this with
N
X
Tr GηN,k = 1
k=0
to obtain (6.36). At this stage we thus have
N
i
h
i 1
h
X
k
(1)
η
0
⊗2 (2) ⊗2
Tr GN,k eH
.
lim inf TrH −∆ηR γN ηR + TrH2 wηR γN ηR ≥ lim
N →∞
N →∞
2
N
k=0
We now use the fundamental relation (5.12) and Corollary 5.4 as indicated in Section 5.3
to deduce
N
N
X
X
k
k
χ
η
0
0
= lim
Tr GN,N −k eH
lim
Tr GN,k eH
N →∞
N →∞
N
N
k=0
k=0
Z
N
X
k
χ
0
= lim
Tr GN,k eH 1 −
e0H (1 − kχR uk2 )dµ(u),
=
N →∞
N
BH
k=0
which concludes the proof of (6.35).
There remains to insert (6.34) and (6.35) in (6.33) and use Fatou’s lemma. This gives
Z h
i
E V (N )
2
V
0
EH [χR u] + eH (1 − kχR uk ) dµ(u)
≥ lim inf
eH (1) ≥ lim
R→∞
N →∞
N
BH
Z
h
i
lim inf EHV [χR u] + e0H (1 − kχR uk2 ) dµ(u)
≥
R→∞
ZBH h
i
2
V
0
EH [u] + eH (1 − kuk ) dµ(u)
=
ZBH h
i
eVH (kuk2 ) + e0H (1 − kuk2 ) dµ(u) ≥ eH (1),
(6.37)
≥
BH
0
eH (λ)
using the continuity of λ 7→
and the binding inequality (6.9). This concludes the
proof of (6.10), and the other results of the theorem follow by inspecting the cases of
equality in (6.37).
90
NICOLAS ROUGERIE
7. Derivation of Gross-Pitaevskii functionals
We now turn to the derivation of non-linear Schrödinger (NLS) functionals with local
non-linearities:
Z
a
|∇ψ|2 + V |ψ|2 + |ψ|4 .
Enls [ψ] :=
2
Rd
One may obtain such objects starting from a Hartree functional such as (3.8) and taking
an interaction potential converging (in the sense of distributions) to a a Dirac mass
Z
x
w δ0 when L → 0.
⇀
wL (x) = L−d w
L
Rd
Since we already obtained (3.8) as the limit of a N -body problem, one might be tempted
to see the derivation of the NLS functional as a simple passage to the limit in a one-body
problem. The issue with such an approach is the total lack of control on the relationship between the physical parameters N and L. One can see this as a problem of non-commuting
limits: it is not clear at all that one can interchange the order of the limits N → ∞ and21
L → 0.
From a physical point of view, the good question is “What relation should N and L
satisfy if one is to obtain the NLS energy by taking simultaneously the limits N → ∞ and
L → 0 of the N -body problem ?” The process leading to the NLS theory is thus more
subtle than that leading to the Hartree theory. We shall first elaborate a little bit more
on this point in the following subsection.
7.1. Preliminary remarks.
Until now we have worked with only two physical parameters: the particle number N
and the interactions strength λ. In order to have a well-defined limit problem we have
been lead to considering the mean-field limit where λ ∝ N −1 . In this case, the range of the
interaction potential is fixed and each particle interacts with all the others. The interaction
strength felt by a typical particle is thus of order λN ∝ 1, comparable with its self-energy
(kinetic + potential). We saw that this equilibration of forces acting on each particle,
combined with structure results à la de Finetti, naturally leads to the conclusion that
particles approximately behave as if they were independent. It follows that Hartree-type
descriptions are valid in the limit N → ∞.
There are other ways to justify such models: the equilibration of forces which allows the
limit problem to emerge may result from a more subtle mechanism. For example, in the
ultra-cold alkali gases where BEC has been observed, it has more to do with the diluteness of the system than with the weakness of the interaction strength. For a theoretical
description of this situation, we can introduce in our model a length scale L characterizing
the range of the interactions. Taking the total system size as a reference, a dilute system
is materialized by taking the limit L → 0. The interaction strength for a typical particle
is then of order λN Ld (interaction strength × number of particles in a ball of radius L
around a given particle). This parameter is the one we should fix when N → ∞ to obtain
a limit problem. Different regimes are then possible, depending on the ratio between λ
and L.
21The L → 0 limit of the N -body problem is by the way very hard to define properly.
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
91
One may discuss the different possibilities starting from the N -body Hamiltonian22
HN =
N
X
j=1
−∆j + V (xj ) +
1
N −1
X
1≤i<j≤N
N dβ w(N β (xi − xj ))
(7.1)
which corresponds to choosing
L = N −β ,
λ ∝ N dβ−1 .
The fixed parameter 0 ≤ β is used to set the ratio between L and λ. We consider the
reference interaction potential w as fixed, and we shall denote
wN (x) := N dβ w(N β x).
For β > 0, wN converges in the sense of distributions to a Dirac mass
Z
w δ0 ,
wN →
(7.2)
(7.3)
Rd
materializing the short range of the interactions/diluteness
of the system. Reasoning
R
formally, one may want to directly replace wN by Rd w δ0 in (7.1). In this case we are
back to a mean-field limit, with a Dirac mass as interaction potential. This is of course
purely formal (except in 1D where the Sobolev injection H 1 ֒→ C 0 allows to properly
define the contact interaction). Accepting that this manipulation has a meaning and that
one may approximate the ground state of (7.1) under the form
ΨN = ψ ⊗N
we obtain the Hartree functional
Z
1
|∇ψ|2 + V |ψ|2 + |ψ|2 (w ∗ |ψ|2 ).
EH [ψ] :=
2
Rd
(7.4)
(7.5)
Replacing the interaction potential by a Dirac mass at the origin leads to the GrossPitaevskii functional
Z
a
|∇ψ|2 + V |ψ|2 + |ψ|4 .
Enls [ψ] :=
(7.6)
2
Rd
In view of (7.3), the logical choice seems to imagine that when β > 0, we obtain, starting
from (7.1), the above functional with
Z
w.
(7.7)
a=
Rd
In fact one may prove the following results (described in the case d = 3; d ≤ 2 leads to
subtleties and d ≥ 4 does not have much physical meaning):
• If β = 0 we have the previously studied mean-field (MF) regime. The range of
the interaction potential is fixed and its strength decreases proportionally to N −1 .
The limit problem is then (7.5), as previously shown.
22Once again, it is possible to add fractional Laplacians and/or magnetic fields, cf Remark 3.2. For
simplicity we do not pursue this here.
92
NICOLAS ROUGERIE
• If 0 < β < 1, the limit problem is (7.6) with the parameter choice (7.7), as can be
expected. We will call this case the non-linear Schrödinger (NLS) limit. This case
does not seem to have been considered in the literature prior to [115] but, at least
when w ≥ 0 and V is confining, one may adapt the analysis of the more difficult
case β = 1.
• If β = 1, the limit functional is now (7.6) with
a = 4π × scattering length of w
(see [125, Appendix C] for a definition). In this case, the ground state of (7.1)
includes a correction to the ansatz (7.4), in the form of short-range correlations.
In fact, one should expect to have
ΨN (x1 , . . . , xN ) ≈
N
Y
j=1
ψ(xj )
Y
1≤i<j≤N
f N β (xi − xj )
(7.8)
where f is linked to the two-body problem defined by w (zero-energy scattering
solution). It so happens that when
R β = 1, the correction has a leading order effect
on the energy, that of replacing Rd w by the corresponding scattering length, as
noted first in [59]. We will call this case the Gross-Pitaevskii (GP) limit. It is
studied in a long and remarkable series of papers by Lieb, Seiringer and Yngvason
(see for example [134, 135, 127, 126, 128, 123, 125]).
As already mentioned, the corresponding evolution problems have also been thoroughly
studied. Here too one has to distinguish the MF limit [10, 5, 76, 165, 157], the NLS
limit [68, 156] and the GP limit [69, 155, 13].
There is a fundamental physical difference between the MF and GP regimes: In both
cases the effective interaction parameter λN L3 is of order 1, but one goes (still in 3D)
from a case with numerous weak collistions when λ = N −1 , L = 1 to a case with rare but
strong collisions when λ = N 2 , L = N −1 . The different NLS scaling sort of interpolate
between these two extremes, the transition from “frequent weak collistions” to “rare strong
collisions” happens at λ = 1, L = N −1/3 , i.e. β = 1/3.
The difficulty for obtaining (7.6) by taking the limit N → ∞ is that there are in fact two
distinct limits N → ∞ and wN → aδ0 to control at the same time. A simple compactness
argument will not suffice in this case and one has to work with quantitative estimates. The
goal of this chapter is to explain how one may proceed starting from the quantitative de
Finetti theorem stated in Chapter 4. Since this theorem is only valid in finite dimension,
we will need a natural way to project the problem onto finite dimensional spaces. We thus
deal with the case of trapped bosons, assuming that for some constants c, C > 0
c|x|s − C ≤ V (x).
(7.9)
In this case, the one-body Hamiltonian −∆ + V has a discrete spectrum on which we have
a good control thanks to Lieb-Thirring-like inequalities.
The results we are going to obtain are valid for 0 < β < β0 where β0 = β0 (d, s) depends
only on the dimension of the configuration space and the potential V . We shall give
explicit estimates of β0 , but the method, introduced in [115], is for the moment limited to
relatively small β. In particular, we will always obtain (7.7) as interaction parameter. The
main advantage compared to the Lieb-Seiringer-Yngvason method [134, 126, 128, 123] is
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
93
that we can in some cases avoid the assumption w ≥ 0, always made in these papers (see
also [125]). In particular we present the first derivation of attractive NLS functionals23 in
1D and 2D.
Recently, a blend of the following arguments, the tools of [134, 126, 128, 123] and new
a priori estimates for the many-body ground state has allowed to extend the analysis to
the GP regime [149]. One can also adapt the method to deal with particles with statistics
slightly deviating from the bosonic one (“almost bosonic anyons”), cf [141].
7.2. Statements and discussion.
We take comfortable assumptions on w:
w ∈ L∞ (Rd , R) and w(x) = w(−x).
(7.10)
Without loss of generality we assume
sup |w| = 1
Rd
to simplify some expressions. We also assume that
x 7→ (1 + |x|)w(x) ∈ L1 (Rd ),
(7.11)
which simplifies the replacement of wN by a Dirac mass. As usual we use the same notation
wN for the interaction potential (7.2) and the multiplication operator by wN (x − y) acting
on L2 (R2d ).
For β = 0 we have shown previously that
E(N )
= eH .
(7.12)
N
We now deal with the case 0 < β < β0 (d, s) < 1 where we obtain the ground state energy
of (7.6):
enls :=
inf
Enls [ψ]
(7.13)
lim
N →∞
kψkL2 (Rd ) =1
with a defined in (7.7). Because of the local non-linearity, the NLS theory is more delicate
than Hartree’s theory. We shall need some structure assumptions on the interactions
potential (see [115] for a more thorough discussion):
• When d = 3, it is well-known that a ground state for (7.6) exists if and only if a ≥ 0.
This is because the cubic non-linearity is super-critical24 in this case. Moreover,
it is easy to see that N −1 E(N ) → −∞ if w is negative at the origin. The optimal
assumption happens to be a classical stability notion for the interaction potential:
ZZ
ρ(x)w(x − y)ρ(y)dxdy ≥ 0, for all ρ ∈ L1 (Rd ), ρ ≥ 0.
(7.14)
Rd ×Rd
This is satisfied as soon as w = w1 + w2 , w1 ≥ 0Rand ŵ2 ≥ 0 with wˆ2 the Fourier
transform of w2 . This assumption clearly implies Rd w ≥ 0, and one may easily see
by changing scales that if it is violated for a certain ρ ≥ 0, then E(N )/N → −∞.
23One often uses the non-linear optics vocabulary to distinguish the repulsive and attractive cases :
repulsive = defocusing, attractive = focusing.
24One may for example consult [102] for a classification of non-linearities in the NLS equation.
94
NICOLAS ROUGERIE
• When d = 2, the cubic non-linearity is critical. A minimizer for (7.6) exists if and
only if a > −a∗ with
a∗ = kQk2L2
(7.15)
where Q ∈ H 1 (R2 ) is the unique [108] (modulo translations) solution to
− ∆Q + Q − Q3 = 0.
The critical interaction parameter
lation inequality
Z
Z
4
|u| ≤ C
a∗
R2
R2
(7.16)
is the best possible constant in the interpo2
|∇u|
Z
2
R2
|u|
.
(7.17)
We refer to [88, 142] for the existence of a ground state and to [196] for the
inequality (7.17). A pedagogical discussion of this kind of subjects may be found
in [74].
R In view ∗of the above conditions, it is clear that in 2D we have to assume
w ≥ −a , but this is in fact not sufficient: as in 3D, if the interaction potential is sufficiently negative at the origin, one may see that N −1 E(N ) → −∞.
The appropriate assumption is now
ZZ
1
2
2
kukL2 k∇ukL2 +
|u(x)|2 |u(y)|2 w(x − y) dx dy > 0
(7.18)
2
R2 ×R2
for all u ∈ H 1 (R2 ). Replacing u by λu(λx) and taking the limit λ → 0 we obtain
Z
Z
1
2
2
|u(x)|4 dx ≥ 0,
∀u ∈ H 1 (R2 ),
w
kukL2 k∇ukL2 +
2
R2
R2
which implies that
Z
R2
w(x) dx ≥ −a∗ .
A scaling argument shows that if the strict inequality in (7.18) is reversed for a
certain u, then E(N )/N → −∞. The case where equality may occur in (7.18) is
left aside in these notes. It requires a more thorough analysis, see [88] where this
is provided at the level of the NLS functional.
• When d = 1, the cubic non-linearity is sub-critical and there is always a minimizer
for the functional (7.6). In this case we need no further assumptions.
We may now state the
Theorem 7.1 (Derivation of NLS ground states).
Assume that either d = 1, or d = 2 and (7.18) holds, or d = 3 and (7.14) holds. Further
suppose that
s
< 1.
(7.19)
0 < β ≤ β0 (d, s) :=
2ds + sd2 + 2d2
where s is the exponent appearing in (7.9). We then have
(1) Convergence of the energy:
E(N )
→ enls .
N
(7.20)
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
95
(2) Convergence of states: Let ΨN be a ground state of (7.1) and
(n)
γN := Trn+1→N [|ΨN ihΨN |]
its reduced density matrices. Along a subsequence we have, for all n ∈ N,
Z
(n)
dµ(u)|u⊗n ihu⊗n |
lim γN =
N →∞
(7.21)
u∈Mnls
strongly in S1 (L2 (Rdn )). Here µ is a probability measure supported on
o
n
Mnls = u ∈ L2 (Rd ), kukL2 = 1, Enls [u] = enls .
(7.22)
In particular, when (7.6) has a unique minimizer unls (up to a constant phase),
we have for the whole sequence
(n)
⊗n
lim γN = |u⊗n
nls ihunls |.
N →∞
(7.23)
Uniqueness of unls is ensured if a ≥ 0 or |a| is small. If these conditions are not
satisfied, one can show that there are several minimizers in certain trapping potentials
having degenerate minima [9, 88].
Remark 7.2 (On the derivation of NLS functionals.).
(1) The assumption β < β0 (d, s) is dictated by the method of proof but it is certainly
not optimal, see [123, 126, 134, 135, 149]. One may relax a bit the condition on β,
at the price of heavier computations, something we prefer to avoid in these notes,
see [115]. In 1D, one may obtain the result for any β > 0.
(2) It is in fact likely that one could replace β0 (d, s) by β(d, ∞) for any s > 0, using
localization methods from Chapter 5. Indeed, ground states always essentially live
in a bounded region. We refrain from pursuing this in order not to add another
layer of localization (in space) to the proof: we will already rely a lot on finite
dimensional localization. In any case, β < β(d, ∞) is not optimal either.
(3) Let us inspect in more details the conditions on β0 (d, s) we obtain. For the case of
a quadratic trapping potential V (x) = |x|2 for example, we can afford β < 1/24 in
3D, β < 1/12 in 2D and β < 1/4 in 1D. The method adapts with no difficulty to
the case of particles in a bounded domain which corresponds to setting formally
s = ∞. We then otain β0 (d, s) = 1/15 in 3D, 1/8 in 2D and 1/3 in 1D. Improving
these thresholds in the case of (partially) attractive potentials remains an open
problem.
(4) When β is smaller than the critical β0 (d, s) by a given amount, the method gives
quantitative estimates for the convergence (7.20), see below. We refer to [115,
Remark 4.2] for a discussion of the cases where a convergence rate for the minimizer
can be deduced, based on tools from [32, 74] and assumptions on the behavior of
the NLS functional.
The proof of this result occupies the rest of the chapter. We proceed in two steps.
The bulk of the analysis consists in obtaining a quantitative estimate of the discrepancy
between the N -body energy per particle N −1 E(N ) and the Hartree energy
eH :=
inf
kukL2 (Rd ) =1
EH [u]
(7.24)
96
NICOLAS ROUGERIE
given by minimizing the functional
ZZ
Z 1
2
2
|u(x)|2 wN (x − y)|u(y)|2 dxdy.
|∇u| + V |u| dx +
EH [u] :=
2
d
d
d
R ×R
R
(7.25)
Theorem 7.3 (Quantitative derivation of Hartree’s theory).
Assume (7.9) and (7.10). Let
1 + 2dβ
.
t :=
2 + d/2 + d/s
If
t > 2dβ,
(7.26)
The latter objects still depend on N when β > 0, whence the necessity to avoid compactness arguments and obtain precise estimates. Once the link between N −1 E(N ) and (7.24)
is established, there remains to estimate the difference |enls − eH |, which is a much easier
task. Most of the restrictive assumptions we have made on w are used only in this second
step. The estimates on the difference |eH − N −1 E(N )| are valid without assuming (7.11)
and (7.14) or (7.18). They thus give some information on the divergence of N −1 E(N ) in
the case where eH does not converge to enls :
(7.27)
then, for all d ≥ 1 there exists a constant Cd > 0 such that
eH ≥
E(N )
≥ eH − Cd N −t+2dβ .
N
(7.28)
Remark 7.4 (Explicit estimates in the mean-field limit).
(1) Condition (7.27) is satisfied if 0 ≤ β < β0 (d, s). For the proof of Theorem 7.1 we
are only interested in cases where |eH | is bounded independently of N , and (7.28)
then gives non-trivial information only if (7.27) holds.
(2) The result is valid in the mean-field case where β = 0 and thus eH does not depend
on N . We then obtain explicit estimates improving on Theorem 3.6. These present
a novelty in the case where Hartee’s functional has several minimizers, or a unique
degenerate minimizer. In other cases25, better estimates are known, with an error
of order N −1 given by Bogoliubov’s theory [117, 176, 86, 56]. See [150] for extensions of Bogoliubov’s theory to cases of mutiple and/or degenerate minimizers.
The proof of Theorem 7.3 occupies Section 7.3. We then complete the proof of Theorem 7.1 in Section 7.4.
7.3. Quantitative estimates for Hartree’s theory.
The main idea of the proof is to apply Theorem 4.1 on a low-energy eigenspace of the
one-body operator
T = −∆ + V
acting on H = L2 (Rd ). Assumption (7.9) ensures that the resolvent of this operator is
compact and thus that its specturm is made of a sequence of eigenvalues tending to infinity.
25The simplest example ensuring uniqueness and non-degeneracy is that where ŵ > 0 with ŵ the Fourier
transform of w.
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
97
We denote P− and P+ the spectral projectors corresponding to energies respectively below
and above a given truncation Λ:
P− = 1(−∞, Λ) (T ) ,
P+ = 1H − P− = P−⊥ .
(7.29)
We also denote
NΛ := dim(P− H) = number of eigenvalues of T smaller than Λ.
(7.30)
Since the precision of the quantitative de Finetti theorem depends on the dimension of
the space on which it applies, it is clearly necessary to have at our disposal a convenient
control of NΛ . The tools to achieve this are well-known under the name of Lieb-Thirring
inequalities, or rather Cwikel-Lieb-Rosenblum in this case. We shall use the following
lemma:
Lemma 7.5 (Number of bound states of a Schrödinger operator).
Let V satisfy (7.9). For all d ≥ 1, there exists a constant Cd > 0 such that, for all Λ large
enough
NΛ ≤ Cd Λd/s+d/2 .
(7.31)
Proof. When d ≥ 3, this is an application of [124, Theorem 4.1]. For d ≤ 2, the result
follows easily by applying [44, Theorem 2.1] or [185, Theorem 15.8], see [115] for details.
The familiarized reader can convince herself that the right side of (7.31) is proportional
to the expected number of energy levels in the semi-classical approximation. We refer
to [124, Chapter 4] for a more thorough discussion of this kind of inequalities.
In the sequel we shall argue as follows:
(1) The eigenvectors of T form an orthogonal basis of L2 (Rd ) on which the N particles
should be distributed. The methods of Chapter 5 provide the right tools to analyze
the repartition of the particles between P− H and P+ H.
(2) If the truncation Λ is chosen large enough, particles living on excited energy levels
will have a much larger energy per unit mass than the Hartree energy we are
aiming at. There can thus only be a small number of particles living on excited
energy levels.
(3) Particles living on P− H form a state on Fs≤N (P− H) (truncated bosonic Fock space
built on P− H). Since P− H has finite dimension, one may use Theorem 4.1 to describe these particles. These will give the Hartree energy, up to an error depending
on Λ and the expected number of P− -localized particles. More precisely, in view
of (4.2), we should expect an error of the form
(Λ + N dβ ) × NΛ
,
N−
(7.32)
i.e. dimension of the localized space × operator norm of the projected Hamiltonian
/ number of localized particles.
(4) We next have to optimize over Λ, keeping the following heuristic in mind: if Λ
is large, there will be many P− -localized particles, which favors the denominator
of (7.32). On the other hand, taking Λ small favors the numerator of (7.32).
Picking an optimal value to balance these two effects leads to the error terms of
Theorem 7.3.
98
NICOLAS ROUGERIE
Proof of Theorem 7.3. The upper bound in (7.28) is as usual proven by taking a trial state
of the form u⊗N . Only the lower bound is non-trivial. We proceed in several steps.
Step 1, truncated Hamiltonian. We first have to convince ourselves that it is legitimate
to think only in terms of P+ and P− -localized particles, as we did above. This is the oject
of the following lemma, which bounds from below the two-body Hamiltonian
H 2 = T ⊗ 1 + 1 ⊗ T + wN
(7.33)
in terms of its P− -localization, P− ⊗ P− H2 P− ⊗ P− and a crude bound on the energy of
the P+ -localized particles.
Lemma 7.6 (Truncated Hamiltonian).
Assume that Λ ≥ CN dβ for a large enough constant C > 0. Then
H2 ≥ P−⊗2 H2 P−⊗2 +
Λ ⊗2 2N 2dβ
P −
4 +
Λ
(7.34)
Proof. We denote
H20 = T ⊗ 1 + 1 ⊗ T
the two-body Hamiltonian with no interactions. We may then write
X
H20 = (P− + P+ )⊗2 H20 (P− + P+ )⊗2 =
Pi ⊗ Pj H20 Pk ⊗ Pℓ
(7.35)
i,j,k,ℓ∈{−,+}
=
X
i,j∈{−,+}
Pi ⊗ Pj H20 Pi ⊗ Pj .
(7.36)
Indeed,
Pi ⊗ Pj H20 Pk ⊗ Pℓ = 0
if i 6= k or j =
6 ℓ, since T commutes with P± and P− P+ = 0. We then note that
which gives
P+ T P+ ≥ ΛP+ and P− T P− ≥ −CP− ,
P+ ⊗ P+ H20 P+ ⊗ P+ ≥ 2Λ P+ ⊗ P+
P+ ⊗ P− H20 P+ ⊗ P− ≥ (Λ − C)P+ ⊗ P−
We conclude
where
P− ⊗ P+ H20 P− ⊗ P+ ≥ (Λ − C)P+ ⊗ P− .
H20 ≥ P−⊗2 H20 P−⊗2 + (Λ − C)Π
Π = P+ ⊗ P+ + P− ⊗ P+ + P+ ⊗ P− = 1H2 − P−⊗2 .
We turn to the interactions:
wN = P−⊗2 + Π wN P−⊗2 + Π .
We have to bound from below the difference
wN − P−⊗2 wN P−⊗2
by controling the off-diagonal terms
ΠwN P−⊗2 + P−⊗2 wN Π
(7.37)
(7.38)
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
99
in (7.38). To this end we write
±
−
+
≥0
with wN
− wN
wN = wN
and we use the well-known fact that the diagonal elements of a self-adjoint operator
control26 the off-diagonal elements.
±
are positive as multiplication operators on L2 (R2d ) we have for all b > 0
Since wN
±
b1/2 P−⊗2 ± b−1/2 Π ≥ 0.
b1/2 P−⊗2 ± b−1/2 Π wN
Combining these inequalities appropriately we obtain
ΠwN P−⊗2 + P−⊗2 wN Π ≥ −bP−⊗2 |wN |P−⊗2 − b−1 Π|wN |Π
for all b > 0. We then recall that, as an operator,
|wN | ≤ kwN kL∞ ≤ N dβ
and we choose b = 2N dβ /Λ to obtain
ΠwN P−⊗2 + P−⊗2 wN Π ≥ −
2N 2dβ ⊗2 Λ
P− − Π
Λ
2
Inserting this bound in (7.38) we get
wN ≥
P−⊗2 wN P−⊗2
2N 2dβ ⊗2
P− −
−
Λ
Λ
dβ
+N
Π
2
(7.39)
by simply bounding from below
ΠwN Π ≥ −Π|wN |Π ≥ −N dβ Π.
Combining with (7.37) and (7.39) we obtain for all Λ ≥ 1 the lower bound
2N 2dβ ⊗2
Λ
⊗2
⊗2
dβ
H2 ≥ P− H2 P− −
P− +
−N −C Π
Λ
2
Since we assume Λ ≥ CN dβ for a large constant C > 0, we may use P− ⊗ P− ≤ 1H2 and
P− ⊗ P+ , P+ ⊗ P− ≥ 0 to deduce
H2 ≥ P−⊗2 H2 P−⊗2 −
Λ
2N 2dβ
+ P+⊗2 ,
Λ
4
which concludes the proof.
Step 2, estimating the localized energy. Let ΨN be a minimizer for the N -body
energy, ΓN = |ΨN ihΨN | and
(n)
γN = Trn+1→N [ΓN ]
the corresponding reduced density matrices. We can now think only in terms of the P−
and P+ -localized states defined as in Lemma 5.2 by the relations
(n)
(n)
(7.40)
G±
= P±⊗n γN P±⊗n .
N
26Cf for a positive hermitian matrix (m )
i,j 1≤i,j≤n the inequality 2|mi,j | ≤ mi,i + mj,j .
100
NICOLAS ROUGERIE
We recall that G±
N are states on the truncated Fock space, i.e.
N
X
TrHk [G±
N,k ] = 1.
(7.41)
k=0
We now compare the Hartree energy eH and the localized energy of ΓN defined by the
truncated Hamiltonian of Lemma 7.6:
Lemma 7.7 (Lower bound for the localized energy).
If Λ ≥ CN dβ for a large enough constant C > 0 we have,
i Λ h
i
ΛNΛ
1 h ⊗2
Λ
(2)
(2)
Tr P− H2 P−⊗2 γN + Tr P+⊗2 γN ≥ eH − C
− C 2.
(7.42)
2
4
N
N
This lemma is proved by combining Theorem 4.1 and the methods of Chapter 5. We
define an approximate de Finetti measure starting from G−
N . The idea is related to that
we used for the proof of the weak de Finetti theorem in Section 5.3:
N −1 E
D
X
N
k
⊗k
du
(7.43)
u⊗k , G−
u
dim (P− H)ks
dµN (u) =
N,k
2
2
k=2
where du is the uniform measure on the finite-dimensional sphere SP− H. The choice of the
weights in the above sum comes from the fact that we want to approximate the localized
(2)
two-body density matrix P−⊗2 γN P−⊗2 , which is the purpose of
Lemma 7.8 (Quantitative de Finetti for a localized state.).
For all Λ > 0, we have
Z
8NΛ
⊗2 (2) ⊗2
⊗2
⊗2
.
|u ihu |dµN (u) ≤
TrH2 P− γN P− −
N
SP− H
G−
N,k
Proof. Up to normalization,
is a state on (P− H)⊗k
s . Applying Theorem 4.1 with the
explicit construction (4.6) we thus have
h
i Z
h
i
NΛ
−
⊗2
⊗2
|u ihu |dµN,k (u) ≤ 8
TrH2 Tr3→k GN,k −
TrHk G−
N,k
k
SP− H
with
E
D
⊗k
du.
dµN,k (u) = dim(P− H)ks u⊗k , G−
N,k u
In view of (7.40) and (7.43) we deduce
Z
N −1 i
h
X
⊗2 (2) ⊗2
N
k 8NΛ
⊗2
⊗2
|u ihu |dµN (u) ≤
TrH2 P− γN P− −
TrHk G−
N,k .
2
2
k
SP− H
k=2
There remains to use the normalization (7.41) and
−1 N
k
k
k(k − 1)
≤
=
N (N − 1)
N
2
2
to conclude the proof.
We proceed to the
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
101
Proof of Lemma 7.7. We start with the P− -localized term. By cyclicity of the trace we
have
i
h
h
i
(2)
(2)
Tr P−⊗2 H2 P−⊗2 γN = Tr P−⊗2 H2 P−⊗2 P−⊗2 γN P−⊗2 .
We then apply Lemma 7.8, which gives
Z
(2)
TrH2 P−⊗2 γN P−⊗2 −
8NΛ
.
|u⊗2 ihu⊗2 |dµN (u) ≤
N
SP− H
On the other hand we of course have
⊗2
P H2 P ⊗2 ≤ 2Λ + kwN kL∞ ≤ 3Λ
−
−
in operator norm, and thus
i 1Z
CΛNΛ
1 h ⊗2
(2)
Tr P− H2 P−⊗2 γN ≥
TrH2 H2 |u⊗2 ihu⊗2 | dµN −
2
2 SP− H
N
Z
CΛNΛ
EH [u]dµN −
=
.
N
SP− H
By the variational principle EH [u] ≥ eH , we deduce
N −1 CΛN
i
X
N
k
1 h ⊗2
Λ
⊗2 (2)
−
TrHk G−
Tr P− H2 P− γN ≥ eH
N,k
2
2
2
N
k=2
R
where the computation of dµN is straightforward using Schur’s formula (4.5).
For the P+ -localized term we use (7.40), (5.10) and (5.12) to obtain
N i
h
Λ X N −1 k
Λ
⊗2 (2) ⊗2
Tr[P+ γN P+ ] =
Tr G+
N,k
4
4
2
2
k=2
h
N −2 i
Λ X N −1 N − k
=
Tr G−
N,k .
4
2
2
(7.44)
(7.45)
(7.46)
(7.47)
k=0
Gathering (7.46), (7.47) and recalling that
−1 −1 N
N −k
(N − k)2
N
k
k2
−1
+ O(N −1 ),
=
= 2 + O(N ),
N
N2
2
2
2
2
we find
i Λ
1 h ⊗2
(2)
(2)
Tr P− H2 P−⊗2 γN + Tr[P+⊗2 γN ]
2
4
N
i k2
h
X
(N − k)2 Λ
C(|eH | + Λ) CΛNΛ
−
Tr GN,k
≥
eH +
−
.
(7.48)
−
2
2
N
N
4
N2
N
k=0
The first error term comes from the fact that the sums in (7.46) and (7.47) do not run
exactly from 0 to N , and we have used the normalization of the localized states (7.41) to
control the missing terms.
It is easy to see that for all p, q, 0 ≤ λ ≤ 1
pλ2 + q(1 − λ)2 ≥ p −
p2
.
q
102
NICOLAS ROUGERIE
We then take p = eH , q = Λ/4, λ = k/N and use (7.41) again to deduce
i Λ
h
(2)
(2)
Tr P−⊗2 H2 P−⊗2 γN + Tr(P+⊗2 γN )
2
e2
C(|eH | + Λ) CΛNΛ
≥ eH − H −
−
.
Λ
N2
N
from (7.48). There remains to insert the simple estimate
|eH | ≤ C + kwN kL∞ ≤ C + N dβ ≤ C + Λ
to obtain the sought-after result.
(7.49)
Step 3, final optimization.. There only remains to optimize over Λ. Indeed, we recall
that by definition
E(N )
1
(2)
= TrH2 [H2 γN ]
N
2
with the two-body Hamiltonian (7.33). Combining Lemmas 7.6 and 7.7 we have the lower
bound
CN 2dβ
ΛNΛ
Λ
E(N )
≥ eH −
−C
−C 2
N
Λ
N
N
for all Λ ≥ CN dβ with C > 0 large enough. Using Lemma 7.5, this reduces to
E(N )
CN 2dβ
Λ1+d/s+d/2
≥ eH −
− Cd
.
N
Λ
N
Optimizing over Λ we get
Λ = Nt
(7.50)
where
1 + 2dβ
t = 2s
4s + ds + 2d
and the condition t > 2dβ in (7.27) ensures that Λ ≫ N 2dβ for large N . We thus conclude
eH ≥
E(N )
≥ eH − Cd N −t+2dβ ,
N
which is the desired result.
Remark 7.9 (Note for later use.).
Following the steps of the proof more precisely we obtain information on the asymptotic
behavior of minimizers. More specifically, going back to (7.45), using Lemma 7.6 and
dropping the positive P+ localized terms we have
Z
1
E(N )
⊗2 (2)
⊗2
EH [u]dµN (u) + o(1)
≥ Tr[P− H2 P− γN ] + o(1) ≥
eH ≥
N
2
SH
and thus
o(1) ≥
Z
SH
(EH [u] − eH ) dµN (u)
(7.51)
when N → ∞. We do not specify here (cf Item (3) in Remark 7.2) the exact order of
magnitude of the o(1) obtained by optimizing in Step 3 above. Estimate (7.51) morally
says that µN must be concentrated close to the minimizers of EH . This will be of use in
the proof of Theorem 7.1.
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
103
7.4. From Hartree to NLS.
There remains to deduce Theorem 7.1 as a corollary of the above analysis. We start
with the following lemma:
Lemma 7.10 (Stability of one-body functionals).
Consider the functionals (7.25) and (7.6). Under the assumptions of Theorem 7.1, there
exists a minimizer for Enls . Moreover, for all normalized function u ∈ L2 (Rd ) we have
k|u|k2H 1 ≤ C(EH [u] + C)
and
Z
|EH [u] − Enls [u]| ≤ CN −β 1 +
Rd
Consequently
(7.52)
|∇u|2
2
.
|eH − enls | ≤ CN −β .
(7.53)
(7.54)
Proof. The stability assumptions we have made guarantee that minimizing sequences for
Enls are bounded in H 1 . Assumption (7.3) allows to easily estimate the difference between
the Hartree and NLS functionals for H 1 functions. Details are omitted and may be found
in [115].
We are now equiped to complete the derivation of NLS functional.
Proof of Theorem 7.1. Combining (7.54) with (7.28) concludes the proof of (7.20). There
thus remains to prove convergence of states, the second item of the theorem. We proceed
in four steps:
Step 1, strong compactness of reduced density matrices. We extract a diagonal
subsequence along which
(n)
(7.55)
γN ⇀∗ γ (n)
when N → ∞, for all n ∈ N. On the other hand we have
i
i
h
h
(1)
(1)
(7.56)
Tr T γN = Tr (−∆ + V ) γN ≤ C,
independently of N . To see this, pick some α > 0, define
HN,α =
N
X
j=1
(−∆j + V (xj )) +
1+α
N −1
X
1≤i<j≤N
N dβ w(N β (xi − xj ))
and apply Theorem 7.3 to this Hamiltonian. We find in particular HN,α ≥ −CN and
deduce
(1) α
hΨN , HN ΨN i
≥ −C(1 + α)−1 +
Tr T γN ,
enls + o(1) ≥
N
1+α
which gives (7.56). Since T = −∆ + V has compact resolvent, (7.55) and (7.56) imply
(1)
that, up to a subsequence γN strongly converges in trace-class norm. As noted previously,
(n)
Theorem 3.5 then implies that also γN strongly converges, for all n ≥ 1.
Step 2, defining the limit measure. We simplify notation by calling rN the best
bound on |E(N )/N − enls | obtained previously. Let dµN be defined as in Lemma 7.8. It
satisfies
i
h
(2)
µN (SP− H) = Tr P−⊗2 γN P−⊗2 .
104
NICOLAS ROUGERIE
We have
Z
⊗2 (2) ⊗2
Tr P− γN P− −
⊗2
SP− H
|u
ihu
⊗2
8NΛ
Λ1+d/s+d/2
≤C
→ 0.
|dµN (u) ≤
N
N
One may on the other hand deduce from the energy estimates of Section 7.3 a control on
the number of excited particles:
i r
h
(2)
N
1 − µN (SP− H) = Tr (1 − P−⊗2 )γN ≤
.
(7.57)
Λ
By the triangle and Cauchy-Schwarz inequalities we deduce
r
Z
(2)
rN
Λ1+d/s+d/2
⊗2
⊗2
+C
.
(7.58)
|u ihu |dµN (u) ≤ C
Tr γN −
N
Λ
SP− H
We now denote PK the spectral projector of T on energies below a truncation K, defined
(2)
as in (7.29). Since γN → γ (2) and PK → 1
lim
lim µN (SPK H) = 1.
K→∞ N →∞
This condition allows us to use Prokhorov’s Theorem and [186, Lemma 1] to ensure that,
after a possible further extracion, µN converges to a measure µ on the ball BH. Passing
to the limit, we find
Z
γ (2) =
BH
|u⊗2 ihu⊗2 |dµ(u)
and it follows that µ has its support in the sphere SH since Tr[γ (2) ] = 1 by strong convergence of the subsequence.
Step 3, the limit measure only charges NLS minimizers. Using (7.51) and
r N
,
µN (SP− H) = 1 + O
Λ
we deduce that
Z
EH [u] − eH dµN (u) ≤ o(1)
SP− H
in the limit N → ∞. By the estimates of Lemma 7.10, it follows that, for a large enough
constant B > 0 independent of N ,
Z
Z
B2
EH [u] − eH dµN (u) ≤ o(1),
dµN (u) ≤
C k∇ukL2 ≥B
k∇ukL2 ≥B
and
Z
k∇ukL2 ≤B
Enls [u] − enls dµN ≤ C(1 + B 4 )N −β +
Z
k∇ukL2 ≤B
EH [u] − eH dµN (u) ≤ o(1).
Passing to the limit N → ∞, we now see that µ is supported in Mnls .
At this stage, using (7.58) and the convergence of µN we have, strongly in trace-class
norm,
Z
(2)
|u⊗2 ihu⊗2 |dµ(u),
γN →
Mnls
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
105
where µ is a probability measure supported in Mnls . Taking a partial trace we obtain
Z
(1)
|uihu|dµ(u)
γN →
Mnls
and there only remains to obtain the convergence of reduced density matrices of order
n > 2.
Step 4, higher order density matrices. We want to obtain
Z
(n)
|u⊗n ihu⊗n |dµ(u),
γN →
Mnls
in trace-class norm when N → ∞. In view of the definition of µ, it suffices to show that
Z
(n)
⊗n
⊗n
|u ihu |dµN (u) → 0
Tr γN −
(7.59)
SP− H
(2)
where µN is the measure defined by applying Lemma 7.8 to γN . To this end we start by
(n)
approximating γN using a new measure, a priori different from µN = µ2N :
N −1 E
D
X
N
k
⊗k
du.
(7.60)
u
dµnN (u) =
dim(P− H)ks u⊗k , G−
N,k
n
n
k=n
Proceeding as in the proof of Lemma 7.8 we obtain
Z
⊗n (n) ⊗n
nNΛ
⊗n
⊗n
n
|u ihu |dµN (u) ≤ C
TrHn P− γN P− −
.
N
(7.61)
SP− H
An estimate similar to (7.57) next shows that
Z
(n)
⊗n
⊗n
n
|u ihu |dµN (u) → 0.
TrHn γN −
SP− H
Using again the bound
−1 n
N
k
k
+ O(N −1 )
=
N
n
n
as well as the triangle inequality and Schur’s formula (4.5) we deduce from (7.61) that
2 n !
Z
N
i
h
X
(n)
k
k
Tr γN −
|u⊗n ihu⊗n |dµN (u) ≤
−
TrHk G−
N,k
N
N
SP− H
k=0
n−1
i
h
X k n
TrHk G−
+
N,k
N
k=0
2 h
i
X
k 2
+
(7.62)
TrHk G−
N,k + o(1).
N
k=0
Finally, combining the various bounds we have obtained
N h
i
X
k 2
TrHk G−
N,k → 1.
N
k=2
106
NICOLAS ROUGERIE
But, because of (7.41) it follows that
N i
h
X
k 2
→ 1.
TrHk G−
N,k
N
k=0
Recalling the normalization
N
X
k=0
h
i
TrHk G−
N,k = 1,
we may thus apply Jensen’s inequality to obtain
!
N N h
i
h
i n/2
X
X
k n
k 2
−
−
1≥
→ 1.
TrHk GN,k ≥
TrHk GN,k
N
N
k=0
k=0
There only remains to insert this and (7.41) in (7.62) to deduce (7.59) and thus conclude
the proof of the theorem.
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
107
Appendix A. A quantum use of the classical theorem
This appendix is devoted to an alternative proof of a weak version of Theorem 3.6. The
method, introduced in [100] is less general than those described previously. This cannot
be helped since it consists in an application of the Hewitt-Savage (classical de Finetti)
theorem to a quantum problem. Here we follow an unpublished note of Mathieu Lewin
and Nicolas Rougerie [118].
In some cases (absence of magnetic fields essentially), the wave-function ΨN minimizing
a N -body energy can be chosen strictly positive. The ground state of the quantum problem may then be entirely analyzed in terms of the N -body density ρΨN = |ΨN |2 , which
is a purely classical object (a symmetric probability measure) whose limit can be described using Theorem 2.1. This approach works only under assumptions on the one-body
Hamiltonian that are much more restrictive than those discussed in Remark 3.2.
A.1. Classical formulation of the quantum problem.
We here consider a quantum N -body Hamiltonian acting on L2 (RdN )
HN =
N
X
(Tj + V (xj )) +
j=1
1
N −1
X
1≤i<j≤N
w(xi − xj )
(A.1)
where the trapping potential V and the interaction potential w are chosen as in Section 3.2.
In particular we assume that V is confining. We shall need rather strong assumptions on
the kinetic energy operator T . The approach we shall discuss in this appendix is based on
the following notion:
Definition A.1 (Kinetic energy with positive kernel).
We say that T has a positive kernel if there exists T (x, y) : Rd × Rd → R+ such that
ZZ
T (x, y) |ψ(x) − ψ(y)|2 dxdy
(A.2)
hψ, T ψi =
for all functions ψ ∈
L2 (Rd ).
Rd ×Rd
It is well-known that the pseudo-relativitic kinetic energy is of this form. Indeed
ZZ
D √
E
Γ( d+1
|ψ(x) − ψ(y)|2
2 )
dxdy,
(A.3)
ψ, −∆ ψ = (n+1)/2
|x − y|d+1
2π
Rd ×Rd
see [122, Theorem 7.12]. More generally one may consider T = |p|s , 0 < s < 2:
ZZ
D
E
|ψ(x) − ψ(y)|2
s
s/2
dxdy
hψ, |p| i = ψ, (−∆) ψ = Cd,s
|x − y|d+s
Rd ×Rd
recalling the correspondance (1.24).
The non-relavistic kinetic energy does not fit in this framework, but one may nevertheless apply the considerations of this appendix to it because
ZZ
Z
|ψ(x) − ψ(y)|2
2
|∇ψ| = Cd lim (1 − s)
hψ, −∆ ψi =
dxdy
(A.4)
s↑1
|x − y|d+2s
Rd ×Rd
Rd
with
Cd =
Z
S d−1
cos θ dσ
−1
.
108
NICOLAS ROUGERIE
Here S d−1 is the euclidean sphere equiped with its Lebesgue measure dσ and θ represents
the angle with respect to the vertical axis. One may thus see −∆ as a limiting case of
Definition A.1. Formula (A.4) is proved in [24, Corollary 2] and [143], see also [25, 144].
The cases with magnetic field T = (p + A)2 and T = |p + A| are not covered by this
framework. One may deal with them with the methods of the main body of the course,
as already mentioned, but not with those of this appendix.
An important consequence of the above choice of kinetic energy is that, by the triangle
inequality
hψ, T ψi ≥ h|ψ|, T |ψ|i
and thus the total N -body energy
EN [ΨN ] = hΨN , HN ΨN i
satisfies
EN [ΨN ] ≥ EN [|ΨN |] .
The ground state energy can thus be calculated using only positive test functions
o
n
o
n
E(N ) = inf EN [ΨN ], ΨN ∈ L2s (RdN ) = inf EN [ΨN ], ΨN ∈ L2s (RdN ), ΨN ≥ 0 . (A.5)
This remark actually allows to prove a fact mentioned previously: the bosonic ground state
is identical to the absolute ground state in the case of a kinetic energy of the form (A.2)
or (A.4), see [124, Chapter 3].
We recall the definition of Hartree’s functional:
ZZ
Z
1
2
|u(x)|2 w(x − y)|u(y)|2 dxdy,
V |u| +
EH [u] = hu, T ui +
2
d
d
d
R ×R
R
its infimum being denoted eH . In the sequel of this appendix we prove the following
statement, which is a weakened version of Theorem 3.6:
Theorem A.2 (Derivation of Hartree’s theory, alternative statement).
Let V and w satisfy the assumptions of Section 3.2, in particular (3.12). We moreover
assume that T has a positive kernel in the sense of Definition A.1, or is a limit case of
this definition, as in (A.4). We then have
E(N )
= eH .
N →∞ N
≥ 0 be a ground state of HN , achieving the infimum in (A.5) and
Z
(n)
|ΨN (x1 , . . . , xN )|2 dxn+1 . . . dxN
ρN (x1 , . . . , xn ) :=
lim
Let ΨN
Rd(N−n)
be its n-body reduced density. There exists a probability measure µ on MH , the set of
minimizers of EH (modulo a phase), such that, along a subsequence
Z
⊗n 2
(n)
u dµ(u) for all n ∈ N,
(A.6)
lim ρN =
L1
dn
N →∞
MH
strongly in
R
. In particular, if eH has a unique minimizer (modulo a constant
phase), then for the whole sequence
2
(n)
for all n ∈ N.
lim ρN = u⊗n
(A.7)
H
N →∞
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
109
Remark A.3 (Uniqueness for Hartree’s theory).
In the case of a kinetic energy with positive kernel, uniqueness of the minimizer of EH is
immediate
p if w ≥ 0. Indeed, the kinetic energy hψ, T ψi is then a strictly convex functional
of ρ = |ψ|2 , see [122, Chapter 7]. If w is positive the functional EH is thus itself strictly
convex as a function of ρ.
The particular case where T = −∆ has been dealt with by Kiessling in [100], and we
shall follow his method in the general case. It consists in treating the problem as a purely
classical one, which explains that we only obtain the convergence of reduced densities (A.6)
instead of the convergence of the full reduced density matrices (3.19). We pursue in the
direction of (A.5) by writing
o
n
√
(A.8)
E(N ) = inf EN [ µN ] , µN ∈ Ps (RdN )
where µN plays the role of |ΨN |2 and we used the fact that we may assume ΨN ≥ 0.
The object we have to study is a symmetric probability measure of N variables, and our
strategy shall be similar to that used to prove Theorem (2.6):
• Since the system is confined, one may easily pass to the limit and obtain a problem
in terms of a classical state with infinitely many particles µ ∈ Ps (RdN ). We
(n)
then use Theorem 2.1 to describe the limit µ(n) of µN , for all n, using a unique
probability measure Pµ ∈ P(P(Rd )).
• The subtle point is to prove that the limit energy is indeed an affine function of µ.
This uses in an essential way the fact that the kinetic energy has a positive kernel
(or is a limit case of such energies), as well as the Hewitt-Savage theorem.
These two steps are contained in the two following sections. We then briefly conclude
the proof of Theorem A.2 in a third section.
A.2. Passing to the limit.
The limit problem we aim at deriving is described by the functional (compare
with (2.50))
q
1
(n)
E[µ] := lim sup T
µ
n→∞ n
ZZ
Z
1
(1)
w(x − y)dµ(2) (x, y), (A.9)
V (x)dµ (x) +
+
2
d
d
d
R ×R
R
where µ ∈ Ps (RdN ) and we set
√
T ( µn ) :=
*
√

µn , 
n
X
j=1
Tj

√
µn
+
(A.10)
for all probability measure µn ∈ P(Rdn ). In fact we shall prove in Lemma (A.5) below
that the lim sup in (A.9) is actually a limit.
Lemma A.4 (Passing to the limit).
Let µN ∈ Ps (RdN ) achieve the infimum in (A.8). Along a subsequence we have
(n)
µN ⇀∗ µ(n) ∈ Ps (Rdn )
110
NICOLAS ROUGERIE
for all n ∈ N, in the sense of measures. The sequence µ(n)
measure µ ∈ Ps (RdN ) and we have
lim inf
N →∞
n∈N
defines a probability
E(N )
≥ E[µ].
N
(A.11)
Proof. Extracting convergent subsequences is done as in Section 2.3. The existence of the
measure µ ∈ Ps (RdN ) follows, using Kolmogorov’s theorem.
Passing to the liminf in the terms
Z
N Z
1 X
(1)
V (x) dµN (x)
V (xj ) dµN (x1 , . . . , xN ) =
N
Rd
RdN
j=1
and
1 1
N N −1
N
X
1≤i<j≤N
Z
RdN
1
w(xi − xj ) dµN (x1 , . . . , xN ) =
2
ZZ
Rd ×Rd
(2)
w(x − y) dµN (x, y)
uses the same ideas as in Chapters 2 and 3. We shall not elaborate on this point.
The new point is to deal with the kinetic energy in order to obtain
q
1
1
√
lim inf T ( µN ) ≥ lim sup T
µ(n) .
(A.12)
N →∞ N
n→∞ n
√
To this end we denote ΨN = µN a minimizer in (A.5) and we then have


N
X
1
1
√
T ( µN ) =
Tr 
Tj |ΨN i hΨN |
N
N
j=1


n
X
1
(n)
Tj γN 
= Tr 
n
j=1
(n)
where γN is the n-th reduced density matrix of ΨN . It is a trace-class operator, that we
decompose under the form
+∞
X
(n)
γN =
λkn |ukn ihukn |
k=1
P
n
with ukn normalised in L2s (Rdk ) and ∞
k=1 λk = 1. Inserting this decomposition in the
previous equation and recalling (A.10) we obtain by linearity of the trace
q
+∞
1
1X k
√
k
2
|un |
T ( µN ) =
λn T
N
n
k=1

v
u +∞
X
u
1
λkn |ukn |2 
≥ T t
n
k=1
q
1
ργ (n)
= T
n
N
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
111
where the inequality in the second line uses the convexity of the kinetic energy as a
functions of the density ρ, already recalled in Remark A.3, cf [122, Chapter 7]. In the last
equality
∞
X
ργ (n) =
λkn |ukn |2
N
is the density
27
of
k=1
(n)
γN
ργ (n)
N
which yields
and it is easy to see that
Z
(n)
|ΨN (x1 , . . . , xN )|2 dxn+1 . . . dxN ,
= ρN =
Rd(N−n)
q
1
1
√
(n)
T ( µN ) ≥ T
µN .
N
n
To obtain (A.12), we first pass to the liminf in N (using Fatou’s lemma), then to the
limsup in n.
The notion of kinetic energy with positive kernel is already crucial at this level. It
provides the convexity property that we just used. It will play an even greater role in the
next section.
A.3. The limit problem. We now have to show that the functional (A.9) is affine on
Ps (RdN ). The last two terms obviously are, which is not suprising since they are classical
in nature. It thus suffices to show that the first term, which encodes the quantum nature
of the problem, is also linear in the density:
Lemma A.5 (Linearity of the limit kinetic energy).
The functional
q
q
1
1
√
(n)
(n)
µ
µ
= lim T
T ( µ) := lim sup T
n→∞ n
n→∞ n
is affine on Ps (RdN ).
Kiessling [100] gave a very elegant proof of this lemma in the case of the non-relativistic
kinetic energy. He notes that in this case
2
q
q
n Z
n Z
2
X
X
1
1
(n) ∇j µ(n) = 1
µ(n) =
log
µ
T
∇j
µ(n)
n
n
4n
dn
dn
R
R
j=1
j=1
and that the last expression is identical to the Fisher information of the probability measure µ(n) . The quantity we study can thus be interpreted as a “mean Fisher information”
of the measure µ ∈ Ps (RdN ), in analogy with the mean entropy introduced in (2.50).
This quantity has an interesting connection to the classical entropy of a probability
measure. Letting µ(n) evolve following the heat flow, one may show that at each time
along the flow, the Fisher information is the derivative of the entropy. Since the heat
flow is linear and the mean entropy is affine (cf Lemma 2.7), Kiessling deduces that the
mean Fisher information is affine. Another point of view on this question is given in [91,
Section 5].
27Formally, the diagonal part of the kernel.
112
NICOLAS ROUGERIE
Here we shall follow a more pedestrian approach whose advantage is to adapt to the
general kinetic energies described in Definition A.1, among other to the pseudo-relativistic
kinetic energy.
Proof. Theorem 2.1 implies that Ps (RdN ) is the convex envelop of the symmetric probability measures characterized by µ(n) = ρ⊗n , ρ ∈ P(Rd ). To prove the lemma it thus suffices
to take
µ1 = ρ⊗n
µ2 = ρ⊗n
µ = 21 µ1 + 12 µ2
1 ,
2 ,
with ρ1 , ρ2 ∈ P(Rd ), ρ1 6= ρ2 and to prove that
q
q
q
(n)
(n) 1
1
(n)
T
µ
µ1
µ2 ≤ o(n) → 0
− 2T
− 2T
(A.13)
when n → ∞. For µi of the above form (i = 1, 2) we clearly have
q
√
(n)
T
µi
= nT ( ρi ) .
Thus the above estimate implies both that the functional under consideration is affine,
and that the lim sup is a limit, given for a general µ ∈ Ps (RdN ) by
q
Z
1
√
(n)
lim T
µ
T ( ρ) dP (ρ)
=
n→∞ n
ρ∈P(Rd )
where P is the classical de Finetti measure of µ defined by Theorem 2.1.
We now prove our claim (A.13). By symmetry of µ(n) and in view of Definition A.1,
we have to compute
q
2
q
ZZ
Z
q
(n)
(n)
(n)
T (x, y) µ (X) − µ (Y ) ,
(A.14)
µ
dX̂
=n
T
Rd(n−1)
Rd ×Rd
where we denote
X = (x1 , . . . , xn ),
Y = (y1 , x2 . . . , xn ),
X̂ = (x2 , . . . , xn ).
We start by claiming that for all X, Y
|I(X, Y )| :=
q
2
2 q
q
2
q
q
q
1
1
(n)
(n)
(n)
(n)
µ(n) (X) − µ(n) (Y ) − µ1 (X) − µ1 (Y ) − µ2 (X) − µ2 (Y ) 2
2
1/2

n
2 p
2 Y
p
p
p
ρ1 (xj )ρ2 (xj )
≤C
ρ1 (x1 ) − ρ1 (y1 ) + ρ2 (x1 ) − ρ2 (y1 ) . (A.15)
j=2
We will prove (A.15) in the case
ρ2 (y1 ) ≤ ρ2 (x1 ) and ρ1 (y1 ) ≤ ρ1 (x1 ).
(A.16)
The adaptation to the other cases is left to the reader. We simplify notation by setting
q
√
ui = ρi , Ui = ρ⊗n
i .
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
113
After expanding the squares we obtain
2I(X, Y ) = U1 (X)U1 (Y ) + U2 (X)U2 (Y )
q
− U12 (X)U12 (Y ) + U22 (X)U22 (Y ) + U12 (X)U22 (Y ) + U22 (X)U12 (Y ).
Then
U12 (X)U12 (Y ) + U22 (X)U22 (Y ) + U12 (X)U22 (Y ) + U22 (X)U12 (Y )
= (U1 (X)U1 (Y ) + U2 (X)U2 (Y ))2 +U12 (X)U22 (Y )+U22 (X)U12 (Y )−2U1 (X)U2 (Y )U1 (Y )U2 (X)
and thus,
2|I(X, Y )| = (U1 (X)U1 (Y ) + U2 (X)U2 (Y ))
s
!
U12 (X)U22 (Y ) + U22 (X)U12 (Y ) − 2U1 (X)U22 (Y )U1 (Y )U2 (X) × 1− 1+
(U1 (X)U1 (Y ) + U2 (X)U2 (Y ))2
U12 (X)U22 (Y ) + U22 (X)U12 (Y ) − 2U1 (X)U22 (Y )U1 (Y )U2 (X)
U1 (X)U1 (Y ) + U2 (X)U2 (Y )


n
Y
|u1 (x1 )u2 (y1 ) − u1 (y1 )u2 (x1 )|2
u1 (xj )u2 (xj )
=
u1 (x1 )u1 (y1 ) + u2 (x1 )u2 (y1 )
j=2


n
Y
|u2 (y1 ) (u1 (x1 ) − u1 (y1 )) + u1 (y1 ) (u2 (y1 ) − u2 (x1 ))|2

u1 (xj )u2 (xj )
=
u1 (x1 )u1 (y1 ) + u2 (x1 )u2 (y1 )
j=2


n
Y
u2 (y1 )2


(u1 (x1 ) − u1 (y1 ))2
u1 (xj )u2 (xj )
≤2
u1 (x1 )u1 (y1 ) + u2 (x1 )u2 (y1 )
j=2
u1 (y1 )2
2
+
(u2 (y1 ) − u2 (x1 )) .
u1 (x1 )u1 (y1 ) + u2 (x1 )u2 (y1 )
≤
Estimate (A.15) immediately follows in the case (A.16) and by similar considerations in
the other cases.
Inserting (A.15) in (A.14) and recalling (A.10) we obtain
q
Z
n−1
q
q
√ √
√
√
(n)
(n) 1
1
(n)
T
µ
µ1
µ2 ≤ Cn
ρ1 ρ2
− 2T
T ( ρ1 )+T ( ρ2 )
− 2T
Rd
(A.17)
√ √ by Fubini. We may assume that T
ρ1 and T
ρ2 are finite, otherwise all the quantities we estimate are equal to +∞ and there is nothing to prove. There remains to note
that
Z
Z
Z
√ √
1
ρ1 ρ2 <
ρ2 < 1
ρ1 +
δ :=
2
Rd
Rd
Rd
since ρ1 6= ρ2 by assumption. We conclude that
q
q
q
(n)
(n)
1
1
(n)
T
− 2T
µ
µ1
µ2 ≤ Cnδn−1
− 2T
and δn−1 → 0 when n → +∞, as desired.
114
NICOLAS ROUGERIE
To deal with the case of the usual, non-relativistic, kinetic energy, we first apply the
preceding arguments to the kinetic energy Ts defined by the positive kernel
Ts (x, y) = Cd |x − y|−(d+2s)
with 0 < s < 1 fixed, to obtain the analogue of (A.17) with T = Ts . We then multiply
this inequality by (1 − s) and take the limit s → 1. Using (A.4) this gives (A.17) with T
the non-relativistic kinetic energy. We may then pass to the limit n → ∞ to obtain the
desired result.
A.4. Conclusion.
The upper bound follows by the usual trial state argument. Combining Lemmas A.4
and A.5 as well as the representation of µ given by Theorem 2.1 we deduce
#
"Z
E(N )
ρ⊗∞ dP (ρ)
=E
lim inf
N →∞
N
P(Rd )
Z
E ρ⊗∞ dP (ρ)
=
P(Rd )
Z
Z
√
EH [ ρ] dµ(ρ) ≥
=
eH dP (ρ) = eH ,
P(Rd )
P(Rd )
where P is the de Finetti measure. This gives the energy convergence. The convergence
of reduced densities follows as usual by noting that equality must hold in all the previous
inequalities.
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
115
Appendix B. Finite-dimensional bosons at large temperature
Until now we have only considered mean-field quantum systems at zero temperature,
and obtained in the limit N → ∞ de Finetti measures concentrated on the minimizers of
the limit energy functional. It is possible, taking a large temperature limit at the same
time as the mean-field limit, to obtain a Gibbs measure. In this appendix we explain this
for the case of bosons with a finite dimensional state-space, following [85, 119] and [104,
Chapter 3].
In infinite dimension, important problems arise, in particular for the definition of the
limit problem. The non-linear Gibbs measures one obtains play an important role in
quantum field theory [54, 181, 193, 77, 81] and in the construction of rough solutions to
the NLS equation, see for example [110, 22, 23, 195, 29, 30, 28, 194, 52]. We refer to the
paper [114] for results on the “mean-field/large temperature” limit in infinite dimension
and a more thorough discussion of these subjects.
B.1. Setting and results.
In this appendix, the one-body state space will be a complex Hilbert space H with finite
dimension
dim H = d.
We consider the mean-field type Hamiltonian
HN =
N
X
hj +
j=1
1
N −1
X
wij
(B.1)
1≤i<j≤N
where h is a self-adjoint operator on H and w a self-adjoint operator on H ⊗ H, symmetric
in the sense that
w(u ⊗ v) = w(v ⊗ u), ∀u, v ∈ H.
The energy functional is as usual defined by
for ΨN ∈
NN
s
EN [ΨN ] = hΨN , HN ΨN i
H and extended to mixed states ΓN over HN =
EN [ΓN ] = TrHN [HN ΓN ] .
NN
s
H by the formula
The equilibrium state of the system at temperature T is obtained by minimizing the
free-energy functional
FN [ΓN ] := EN [ΓN ] + T Tr [ΓN log ΓN ]
amongst mixed states. The minimizer is the Gibbs state
exp −T −1 HN
.
ΓN =
Tr [exp (−T −1 HN )]
(B.2)
(B.3)
The associated minimum free-energy is obtained from the partition function (normalisation
factor in (B.3)) as follows:
1
N
FN = inf FN [ΓN ], ΓN ∈ S(H ) = −T log Tr exp − HN
.
(B.4)
T
We shall be interested in the limit of these objects in the limit
N → ∞,
T = tN,
t fixed
(B.5)
116
NICOLAS ROUGERIE
which happens to be the good regime to obtain an interesting limit problem. We will in
fact obtain a classical free-energy functional that we now define.
Since H is finite-dimensional, one may define du, the normalized Lebesgue measure on
its unit sphere SH. The limiting objects will be de Finetti measures, hence probability
measures µ on SH, and more precisely functions of L1 (SH, du). We introduce for these
objects a classical free-energy functional
Z
Z
µ(u) log (µ(u)) du
(B.6)
EH [u]µ(u)du + t
Fcl [µ] =
SH
SH
and denote Fcl its infimum amongst positive normalized L1 functions. It is attained by
the Gibbs measure
exp −t−1 EH [u]
(B.7)
µcl = R
−1
SH exp (−t EH [u]) du
and we have
Fcl = −t log
Here EH [u] is the Hartree functional
EH [u] =
Z
1
exp − EH [u] du
t
SH
1 ⊗N
u , HN u⊗N HN = hu, huiH + 21 hu ⊗ u, wu ⊗ uiH2 .
N
(B.8)
The theorem we shall prove, due to Gottlieb [85] (see also [84, 95] and, independently [104,
Chapter 3]) is of a semi-classical nature since it provides a link between the quantum and
classical Gibbs states, (B.3) and (B.7):
Theorem B.1 (Mean-field/large temperature limit in finite dimension).
In the limit (B.5), we have
FN = −T log dim HN
s + N Fcl + O(d).
(B.9)
(n)
Moreover, denoting γN the n-th reduced density matrix of the Gibbs state (B.3),
Z
(n)
|u⊗n ihu⊗n |µcl (u)du
(B.10)
γN →
SH
strongly in the trace-class norm of S1 (Hn ).
Remark B.2 (Mean-field/large temperature limit).
A few comments:
(1) One should understand this theorem as saying that essentially, in the limit under
consideration,
Z
|u⊗N ihu⊗N | µcl (u)du.
ΓN ≈
SH
The Gibbs state is thus close to a superposition of Hartree states. The notions of
reduced density matrices and de Finetti measures provide the appropriate manner
to make this rigorous. We will see that the de Finetti measure (lower symbol)
associated to ΓN by the methods of Chapter 4 converges to µcl (u)du.
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
117
(2) Note that the first term in the energy expansion (B.9) diverges very rapidly,
see (4.17). The classical free-energy only appears as a correction. In view of the
dependence on d of this first term, it is clear that the approach in this appendix
cannot be adapted easily to an infinite dimensional setting.
(3) Our method of proof differs from that of [85]. We shall exploit more fully the semiclassical nature of the problem by using the Berezin-Lieb inequalities introduced
in [16, 120, 184]. The method we present [119] owes a lot to the seminal paper [120]
and is reminiscent of some aspects of [128].
(4) It will be crucial for the proof that the lower symbol of ΓN is an approximate de
Finetti measure for ΓN . This will allow us to apply the first Berezin-Lieb inequality
to obtain a lower bound to the entropy term. A new interest of the constructions
of Chapter 4 is thus apparent here. We use not only the estimate of Theorem 4.1
but also the particular form of the constructed measure.
B.2. Berezin-Lieb inequalities.
We recall the resolution of the identity (4.5) over HN given by Schur’s lemma. We thus
have for each state ΓN ∈ S(HN
s ) a lower symbol defined as
⊗N
µN (u) = dim HN
ihu⊗N | .
s Tr ΓN |u
The first Berezin-Lieb inequality is the following statement:
Lemma B.3 (First Berezin-Lieb inequality).
+
Let ΓN ∈ S(HN
s ) have lower symbol µN and f : R → R be a convex function. We have
Z
µN
N
du.
(B.11)
f
Tr [f (ΓN )] ≥ dim Hs
dim (HN
SH
s )
The second Berezin-Lieb inequality applies to states having a positive upper symbol
(see Section 4.2). One may in fact show that every state has an upper symbol, but it is
in general not a positive measure.
Lemma B.4 (Second Berezin-Lieb inequality).
Let ΓN ∈ S(HN
s ) have upper symbol µN ≥ 0,
Z
|u⊗N ihu⊗N |µN (u)du
ΓN =
(B.12)
u∈SH
and f : R+ → R be a convex function. We have
Z
N
Tr [f (ΓN )] ≤ dim Hs
f
SH
µN
dim (HN
s )
du.
(B.13)
Proof of Lemmas B.3 and B.4. We follow [184]. Since ΓN is a state we decompose it under
the form
∞
X
ΓN =
λkN |VNk ihVNk |
with
VNk
∈
HN
s
normalized and
P
k=1
k
k λN
= 1. We denote
D
E2
µkN (u) = VNk , u⊗N 118
NICOLAS ROUGERIE
and by (4.5) we have
dim HN
s
Z
SH
E
D
µkN (u)du = VNk , VNk = 1.
On the other hand, since (VNk )k is a basis of HN
s , for all u ∈ SH
E2
X
X D
µkN (u) =
VNk , u⊗N = 1.
k
(B.14)
(B.15)
k
First inequality. Here µN is the lower symbol of ΓN and we have
X k k
µN (u) = dim HN
λN µN (u)
s
k
and thus
dim
HN
s
Z
f
SH
µN
dim (HN
s )
HN
s
du ≤ dim
Z
X f λkN µkN (u)du
SH k
by Jensen’s inequality and (B.15). Next
Z X X N
dim Hs
f λkN µkN (u)du =
f λkN = Tr [f (ΓN )]
SH k
k
using (B.14).
Second inequality. Here ΓN and µN are related via (B.12). We write
X X Tr [f (ΓN )] =
f λkN =
f hVNk , ΓN VNk i
k
k
X Z
=
f
k
SH
µN (u)µkN (u)du
µN (u)
≤
µkN (u)du
dim
f
N)
dim
(H
SH
s
k
Z
µN (u)
du
= dim HN
f
s
dim (HN
SH
s )
X
HN
s
Z
using Jensen’s inequality and (B.14) to prove the inequality in the third line and
then (B.15) to conclude.
We have here presented a specific version of these general inequalities. It is clear that
the proof applies more generally to any self-adjoint operator on a separable Hilbert space
having a coherent state decomposition of the form (4.5). In the next section these inequalities will be used to deal with the entropy term by taking f (x) = x log x. This will complete
the treatment of the energy using Theorem 4.1 and make the link with the discussion of
Chapter 4.
DE FINETTI THEOREMS AND BOSE-EINSTEIN CONDENSATION
119
B.3. Proof of Theorem B.1.
Upper bound. We take as a trial state
Z
|u⊗N ihu⊗N | µcl (u)du.
Γtest
:=
N
SH
The energy being linear in the density matrix we have
Z
Z
⊗N ⊗N test EN |u ihu | µcl (u)du = N
EN Γ N =
SH
SH
EH [u]µcl (u)du.
For the entropy term we use the second Berezin-Lieb inequality, Lemma B.4, with f (x) =
x log x. This gives
Z
test
µcl (u)
µcl (u)
test
N
Tr ΓN log ΓN ≤ dim Hs
log
du
N
dim (HN
SH dim (Hs )
s )
Z
µcl (u) log (µcl (u)) du.
= − log dim HN
s +
SH
Summing these estimates we obtain
FN ≤ FN Γtest
≤ −T log dim HN
N
s + N Fcl
since µcl minimizes Fcl .
Lower bound. For the energy we use the reduced density matrices as usual to write
i
h
i N
h
(2)
(1)
TrH2 wγN .
EN [ΓN ] = N TrH hγN +
2
Denoting
⊗N
µN (u) = dim HN
u , ΓN u⊗N
s
the lower symbol of ΓN , we recall that it has been proved in Chapter 4 that
Z
(1)
d
TrH γN −
|uihu|µN (u)du ≤ C1
N
SH
Z
(2)
d
|u⊗2 ihu⊗2 |µN (u)du ≤ C2 .
TrH2 γN −
N
SH
Since we work in finite dimension, h and w are bounded operators and it follows that
Z
Z
N
TrH2 w|u⊗2 ihu⊗2 | µN (u)du − Cd
TrH [h|uihu|] µN (u)du +
EN [ΓN ] ≥ N
2 SH
ZSH
EH [u]µN (u)du − Cd.
=N
SH
To estimate the entropy we use the first Berezin-Lieb inequality, Lemma B.3, with f (x) =
x log x. This yields
Z
µN (u)
µN (u)
N
Tr [ΓN log ΓN ] ≥ dim Hs
log
du
N
dim (HN
SH dim (Hs )
s )
Z
µN (u) log (µN (u)) du.
= − log dim HN
s +
SH
120
NICOLAS ROUGERIE
There only remains to sum these estimates to deduce
FN = FN [ΓN ] ≥ −T log dim HN
s + N Fcl [µN ] − Cd
≥ −T log dim HN
s + N Fcl − Cd
since
Z
µN (u)du = 1
SH
by definition.
Convergence of reduced density matrices. The lower symbol µN (u)du is a probability
measure on the compact space SH. We extract a converging subsequence
µN (u)du → µ(du) ∈ P(SH)
and it follows from the results of Chapter 4 that, for all n ≥ 0, along a subsequence,
Z
(n)
|u⊗n ihu⊗n |dµ(u).
(B.16)
γN →
SH
Combining the previous estimates gives
d
.
(B.17)
N
To pass to the liminf N → ∞ in this inequality, the energy term is dealt with as in the
preceding chapters. For the entropy term we note that since du is normalized
Z
µN log µN du ≥ 0
Fcl ≥ Fcl [µN ] − C
SH
since this quantity may be interpreted as the relative entropy of µN with respect to the
constant function 1. Using Fatou’s lemma we thus deduce from (B.17) that
Fcl ≥ Fcl [µ]
and thus dµ(u) = µcl (u)du by uniqueness of the minimizer of Fcl . Uniqueness of the
limit also guarantees that the whole sequence converges and there only remains to go back
to (B.16) to conclude the proof.
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